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" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)*(c+d) =a*c+a*d+b*c+b*d" "6#/*&,&%\"aG\"\"\"%\"bGF'F',&%\"cGF'%\"dGF'F',**&F& F'F*F'F'*&F&F'F+F'F'*&F(F'F*F'F'*&F(F'F+F'F'" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Each term in the second bracket is multiplied by each term in the first bracket . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 400 98 98 {PLOTDATA 2 "6C-%%TEXTG6$7$$\"\"\"\"\"!$F)F)Q\"(6\"-F$6$7$$\"\"#F)F*Q\"aF,-F$6$7 $$\"\"$F)F*Q\"+F,-F$6$7$$\"\"%F)F*Q\"bF,-F$6$7$$\"\"&F)F*Q\")F,-F$6$7$ $\"\"'F)F*F+-F$6$7$$\"\"(F)F*Q\"cF,-F$6$7$$\"\")F)F*F8-F$6$7$$\"\"*F)F *Q\"dF,-F$6$7$$\"#5F)F*FD-F$6$7$$\"#6F)F*Q\"~F,-F$6$7$$\"#7F)F*Q\"=F,- F$6$7$$\"#8F)F*F_o-F$6$7$$\"#9F)F*Q$a~cF,-F$6$7$$\"#:F)F*F_o-F$6$7$$\" #;F)F*F8-F$6$7$$\"#F)F*F_o- F$6$7$$\"#?F)F*F8-F$6$7$$\"#@F)F*F_o-F$6$7$$\"#AF)F*Q$b~cF,-F$6$7$$\"# BF)F*F_o-F$6$7$$\"#CF)F*F8-F$6$7$$\"#DF)F*F_o-F$6$7$$\"#EF)F*Q$b~dF,-% 'CURVESG6%7)7$$\"3%)************p?!#F--F$6$7$7$F($F2F)7$F+FFF--F$6$7$7$F9F(7$F9F9F--%%TEXTG6%7$$\"\"#F )$\"#DF3Q*area~=~ac6\"-%'COLOURG6&F0F(F($\"*++++\"!\")-FN6%7$FQ$\"\"&F 3Q*area~=~bcFVFW-FN6%7$$F[oF)FSQ*area~=~adFVFW-FN6%7$F`oFjnQ*area~=~bd FVFW-FN6$7$$!#:!\"#FjnQ\"bFV-FN6$7$FioFSQ\"aFV-FN6$7$FQ$\"$=%F[pQ\"cFV -FN6$7$F`oFdpQ\"dFV-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6$Q!FVFbq-%%VIE WG6$%(DEFAULTGFfq" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "We \+ can also expand powers such as " }{XPPEDIT 18 0 "(a+b)^2;" "6#*$,&%\"a G\"\"\"%\"bGF&\"\"#" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(a+b)^2=(a+b)*(a+b)" "6#/*$,&%\"aG\"\"\"%\"bGF'\"\"# *&,&F&F'F(F'F',&F&F'F(F'F'" }{XPPEDIT 18 0 "``=a^2+a*b+b*a+b^2" "6#/%! G,**$%\"aG\"\"#\"\"\"*&F'F)%\"bGF)F)*&F+F)F'F)F)*$F+F(F)" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "``=a^2+2*a*b+b^2" "6#/%!G,(*$% \"aG\"\"#\"\"\"*(F(F)F'F)%\"bGF)F)*$F+F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "The multiplicat ion needed to construct this expansion can be set out as follows." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a+b" "6#,&%\"aG\"\"\"%\"bGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+b" "6#,&%\"aG\"\"\"%\"bGF %" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 23 " ____________ \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a*b + b^2" " 6#,&*&%\"aG\"\"\"%\"bGF&F&*$F'\"\"#F&" }{TEXT -1 3 " " }}{PARA 256 " " 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "a^2+a*b" "6#,&*$%\"aG\"\"#\" \"\"*&F%F'%\"bGF'F'" }{TEXT -1 23 " " }}{PARA 256 "" 0 "" {TEXT -1 25 " ____________ " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2+2*a*b+b^2" "6#,(*$%\"aG\"\"#\"\" \"*(F&F'F%F'%\"bGF'F'*$F)F&F'" }{TEXT -1 11 " " }}{PARA 0 " " 0 "" {TEXT -1 30 "Now consider the expansion of " }{XPPEDIT 18 0 "(a +b)^3" "6#*$,&%\"aG\"\"\"%\"bGF&\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^3;" "6#*$,&%\"aG\"\"\"%\"bGF &\"\"$" }{TEXT -1 20 " can be obtained as " }{XPPEDIT 18 0 "(a+b)*(a+b )^2 = (a+b)*(a^2+2*a*b+b^2);" "6#/*&,&%\"aG\"\"\"%\"bGF'F'*$,&F&F'F(F' \"\"#F'*&,&F&F'F(F'F',(*$F&F+F'*(F+F'F&F'F(F'F'*$F(F+F'F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 129 "We need to form and add togethe r six terms, three that arise by multiplying each of the three terms i n the second bracket by the " }{TEXT 264 1 "a" }{TEXT -1 114 " from th e first bracket, and three that arise by multiplying each of the three terms in the second bracket by the " }{TEXT 265 1 "b" }{TEXT -1 72 " \+ from the first bracket. This multiplication can be set out as follows. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a^2+2*a*b+b^2" "6#,(*$%\"aG\"\"#\"\"\"*(F&F'F%F'%\"b GF'F'*$F)F&F'" }{TEXT -1 13 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+b" "6#,&%\"aG\"\"\"%\"bGF%" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 53 " ______________________ \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*b+2*a*b^2+b^3" "6#,(*&%\"aG\"\"#%\"bG\"\"\"F(*(F&F(F%F(F'F&F(*$F' \"\"$F(" }{TEXT -1 21 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^3+2*a^2*b+a*b^2" "6#,(*$%\"aG\"\"$\" \"\"*(\"\"#F'*$F%F)F'%\"bGF'F'*&F%F'*$F+F)F'F'" }{TEXT -1 41 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 54 " _ _____________________ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^3+3*a^2*b+3*a*b^2+b^3" "6#,**$% \"aG\"\"$\"\"\"*(F&F'*$F%\"\"#F'%\"bGF'F'*(F&F'F%F'F+F*F'*$F+F&F'" } {TEXT -1 31 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "This shows that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^3=a^3+3*a^2*b+3*a*b^2+b^3 " "6#/*$,&%\"aG\"\"\"%\"bGF'\"\"$,**$F&F)F'*(F)F'*$F&\"\"#F'F(F'F'*(F) F'F&F'F(F.F'*$F(F)F'" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 266 20 "____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "( a+b)^4;" "6#*$,&%\"aG\"\"\"%\"bGF&\"\"%" }{TEXT -1 20 " can be obtaine d as " }{XPPEDIT 18 0 "(a+b)*(a+b)^3 = (a+b)*(a^3+3*a^2*b+3*a*b^2+b^3) ;" "6#/*&,&%\"aG\"\"\"%\"bGF'F'*$,&F&F'F(F'\"\"$F'*&,&F&F'F(F'F',**$F& F+F'*(F+F'*$F&\"\"#F'F(F'F'*(F+F'F&F'F(F2F'*$F(F+F'F'" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 46 "The multiplication can be set out as \+ follows. " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a^3+3*a ^2*b+3*a*b^2+b^3" "6#,**$%\"aG\"\"$\"\"\"*(F&F'*$F%\"\"#F'%\"bGF'F'*(F &F'F%F'F+F*F'*$F+F&F'" }{TEXT -1 31 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a+b" "6#,&%\"aG\"\" \"%\"bGF%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 62 " _________ _______________ " }}{PARA 256 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "a^3*b+3*a^2*b^2+3*a*b^3+b^4; " "6#,**&%\"aG\"\"$%\"bG\"\"\"F(*(F&F(*$F%\"\"#F(F'F+F(*(F&F(F%F(F'F&F (*$F'\"\"%F(" }{TEXT -1 43 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^4+3*a^3*b+3*a^ 2*b^2+a*b^3" "6#,**$%\"aG\"\"%\"\"\"*(\"\"$F'*$F%F)F'%\"bGF'F'*(F)F'*$ F%\"\"#F'F+F.F'*&F%F'*$F+F)F'F'" }{TEXT -1 59 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 75 "______________________________ \+ " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "a^4+4*a^3*b+6*a^2*b^2+4*a*b ^3+b^4;" "6#,,*$%\"aG\"\"%\"\"\"*(F&F'*$F%\"\"$F'%\"bGF'F'*(\"\"'F'*$F %\"\"#F'F+F/F'*(F&F'F%F'F+F*F'*$F+F&F'" }{TEXT -1 48 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "Each of the five terms has the form " } {XPPEDIT 18 0 "C*a^i*b^j" "6#*(%\"CG\"\"\")%\"aG%\"iGF%)%\"bG%\"jGF%" }{TEXT -1 8 ", where " }{TEXT 267 1 "C" }{TEXT -1 74 " is the numerica l coefficient of the term. In a particular term the power " }{TEXT 268 1 "i" }{TEXT -1 4 " of " }{TEXT 270 1 "a" }{TEXT -1 15 " and the p ower " }{TEXT 269 1 "j" }{TEXT -1 4 " of " }{TEXT 271 1 "b" }{TEXT -1 64 " always have the sum 4. This includes the possibility of either " }{TEXT 272 1 "i" }{TEXT -1 4 " or " }{TEXT 273 1 "j" }{TEXT -1 28 " be ing 0. In the first term " }{XPPEDIT 18 0 "a^4" "6#*$%\"aG\"\"%" } {TEXT -1 2 ", " }{TEXT 274 1 "i" }{TEXT -1 12 " is 4 while " }{TEXT 275 1 "j" }{TEXT -1 76 " is 0. In fact, when proceding from one term t o the next term, the power of " }{TEXT 276 1 "a" }{TEXT -1 35 " decrea ses by 1 while the power of " }{TEXT 277 1 "b" }{TEXT -1 17 " increase s by 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Looking at the coefficients alone in the previous multiplication w e have the following pattern. " }}{PARA 256 "" 0 "" {TEXT -1 40 " \+ 1 3 3 1 " }}{PARA 256 "" 0 "" {TEXT -1 40 " 1 3 3 1 " }}{PARA 256 "" 0 "" {TEXT -1 24 " ______________________ " }}{PARA 256 "" 0 "" {TEXT -1 41 " 1 \+ 4 6 4 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 139 "Instead of writing the line of coeff icients 1, 3, 3, 1 twice, we can generate the new line of coefficien ts by using the addition scheme: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 102 84 84 {PLOTDATA 2 "6)-%'CURVESG6'7$7$$\"\"&!\"\"$F)\"\" !7$$\"#:F*F+7%7$$\"++++q7!\"*$\"+++++`F4F-7$F2$\"+++++ZF4-%&STYLEG6#%, PATCHNOGRIDG-%'COLOURG6&%$RGBGF,F,F,-%*THICKNESSG6#\"\"\"-F$6'7$7$$\" \"#F,$\"#OF*7$FJ$\"#5F*7%7$$\"+++++@F4$\"++++)f\"F4FN7$$\"+++++>F4FUF: F>FB-%%TEXTG6%7$$F,F,F+Q\"p6\"-%%FONTG6$%*HELVETICAGFP-Fen6%7$FJF+Q\"q FjnF[o-Fen6%7$FJFhnQ&p~+~qFjnF[o-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;Fhn$ \"#DF*;$F*F,$\"#bF*" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 186 "applied to the single r ow 1, 3, 3, 1. If there is no number in a position where one is requ ired to complete the configuration above, then a 0 is placed (or imagi ned) in that position. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 470 115 115 {PLOTDATA 2 "6B-%'CURVESG6'7$7$$\"#:!\"\"$\"\"% \"\"!7$$\"#DF*F+7%7$$\"+++++B!\"*$\"++++]UF5F.7$F3$\"++++]PF5-%&STYLEG 6#%,PATCHNOGRIDG-%&COLORG6&%$RGBG$F-F-$\"\")F*FC-%*THICKNESSG6#\"\"#-F $6'7$7$$\"#NF*F+7$$\"#XF*F+7%7$$\"+++++VF5F6FP7$FUF9F;-F@6&FB$F,F*FCFD FF-F$6'7$7$$\"#bF*F+7$$\"#lF*F+7%7$$\"+++++jF5F6F[o7$F`oF9F;-F@6&FB$\" \"\"F-$FfoF*FCFF-F$6'7$7$$\"#vF*F+7$$\"#&)F*F+7%7$$\"+++++$)F5F6F^p7$F cpF9F;-F@6&FB$\"\"(F*FhpFCFF-F$6'7$7$$\"#&*F*F+7$$\"$0\"F*F+7%7$$\"+++ +I5!\")F6F`q7$FeqF9F;-F@6&FB$F]o!\"#$FIF*F]rFF-F$6'7$7$$\"\"$F-Fbr7$Fb rFeo7%7$$\"++++DJF5$\"+++++9F5Fdr7$$\"++++vGF5FirF;F?FF-F$6'7$7$$\"\"& F-Fbr7$FbsFeo7%7$$\"++++D^F5FirFds7$$\"++++v[F5FirF;FXFF-F$6'7$7$$FipF -Fbr7$F`tFeo7%7$$\"++++DrF5FirFat7$$\"++++voF5FirF;FcoFF-F$6'7$7$$\"\" *F-Fbr7$F]uFeo7%7$$\"++++D\"*F5FirF_u7$$\"++++v))F5FirF;FfpFF-F$6'7$7$ $\"#6F-Fbr7$F[vFeo7%7$$\"+++]76FgqFirF]v7$$\"+++](3\"FgqFirF;FiqFF-%%T EXTG6$7$FeoF+Q\"06\"-Ffv6$7$$FIF-F+Q\"~Fjv-Ffv6$7$FbrF+Q\"1Fjv-Ffv6$7$ F+F+F_w-Ffv6$7$FbsF+Q\"3Fjv-Ffv6$7$$\"\"'F-F+F_w-Ffv6$7$F`tF+Fjw-Ffv6$ 7$$FEF-F+F_w-Ffv6$7$F]uF+Fcw-Ffv6$7$$\"#5F-F+F_w-Ffv6$7$F[vF+Fiv-Ffv6$ 7$FbrFCFcw-Ffv6$7$F+FCF_w-Ffv6$7$FbsFCQ\"4Fjv-Ffv6$7$F^xFCF_w-Ffv6$7$F `tFCQ\"6Fjv-Ffv6$7$FfxFCF_w-Ffv6$7$F]uFCF[z-Ffv6$7$F]yFCF_w-Ffv6$7$F[v FCFcw-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;FC$\"#7F-;$F*F-Fbs" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16 " "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "C urve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "We cou ld just display the two rows as follows. " }}{PARA 256 "" 0 "" {TEXT -1 30 " 1 3 3 1 " }}{PARA 256 "" 0 "" {TEXT -1 51 " 1 4 6 4 1 " }}{PARA 0 "" 0 "" {TEXT -1 37 "The coefficients in the expansion of " }{XPPEDIT 18 0 "(a+b)^5" "6#*$,&%\"aG\"\"\"%\"bGF&\"\"&" }{TEXT -1 34 " can be obta ined in a similar way." }}{PARA 256 "" 0 "" {TEXT -1 60 " 0 1 \+ 4 6 4 1 0 " }}{PARA 256 "" 0 "" {TEXT -1 59 " 1 5 10 10 5 \+ 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The expansion of " }{XPPEDIT 18 0 "(a+b)^5" "6#*$,&%\"aG\"\"\"%\"bGF&\"\" &" }{TEXT -1 23 " has terms of the form " }{XPPEDIT 18 0 "C*a^i*b^j" " 6#*(%\"CG\"\"\")%\"aG%\"iGF%)%\"bG%\"jGF%" }{TEXT -1 7 " where " } {XPPEDIT 18 0 "i+j=5" "6#/,&%\"iG\"\"\"%\"jGF&\"\"&" }{TEXT -1 16 ". W e start with " }{XPPEDIT 18 0 "a^5*b^0=a^5" "6#/*&%\"aG\"\"&%\"bG\"\"! *$F%F&" }{TEXT -1 44 " multiplied by the first coefficient 1 from " } {XPPEDIT 18 0 "1,5,10,10,5,1" "6(\"\"\"\"\"&\"#5F%F$F#" }{TEXT -1 26 " and then add the term in " }{XPPEDIT 18 0 "a^4*b^1=a^4*b" "6#/*&%\"aG \"\"%%\"bG\"\"\"*&F%F&F'F(" }{TEXT -1 52 " multiplied by the second co efficient 5, and so on. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The expansion is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^5=a^5+ 5*a^4*b+10*a^3*b^2+10*a^2*b^3+5*a*b^4+b^5" "6#/*$,&%\"aG\"\"\"%\"bGF' \"\"&,.*$F&F)F'*(F)F'*$F&\"\"%F'F(F'F'*(\"#5F'*$F&\"\"$F'F(\"\"#F'*(F0 F'*$F&F3F'F(F2F'*(F)F'F&F'F(F.F'*$F(F)F'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "Starting with the coefficients \"" }{XPPEDIT 18 0 "1,1" "6$\"\"\"F#" }{TEXT -1 5 "\" of " }{XPPEDIT 18 0 "a + b" "6#,&%\"aG\"\"\"%\"bGF%" }{TEXT -1 6 " and \+ \"" }{XPPEDIT 18 0 "1, 2, 1" "6%\"\"\"\"\"#F#" }{TEXT -1 5 "\" of " } {XPPEDIT 18 0 "(a+b)^2 = a^2+2*a*b+b^2;" "6#/*$,&%\"aG\"\"\"%\"bGF'\" \"#,(*$F&F)F'*(F)F'F&F'F(F'F'*$F(F)F'" }{TEXT -1 17 " we can generate \+ " }{TEXT 259 17 "Pascal's triangle" }{TEXT -1 29 ". using the addition scheme: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 102 84 84 {PLOTDATA 2 "6)-%'CURVESG6'7$7$$\"\"&!\"\"$F)\"\"!7$$\"#:F*F+7%7$$\"++ ++q7!\"*$\"+++++`F4F-7$F2$\"+++++ZF4-%&STYLEG6#%,PATCHNOGRIDG-%'COLOUR G6&%$RGBGF,F,F,-%*THICKNESSG6#\"\"\"-F$6'7$7$$\"\"#F,$\"#OF*7$FJ$\"#5F *7%7$$\"+++++@F4$\"++++)f\"F4FN7$$\"+++++>F4FUF:F>FB-%%TEXTG6%7$$F,F,F +Q\"p6\"-%%FONTG6$%*HELVETICAGFP-Fen6%7$FJF+Q\"qFjnF[o-Fen6%7$FJFhnQ&p ~+~qFjnF[o-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;Fhn$\"#DF*;$F*F,$\"#bF*" 1 2 0 1 10 0 2 9 1 1 2 1.000000 46.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 71 " \+ 1 1 " }}{PARA 0 "" 0 "" {TEXT -1 79 " \+ 1 2 1 " }}{PARA 0 " " 0 "" {TEXT -1 88 " \+ 1 3 3 1 " }}{PARA 0 "" 0 "" {TEXT -1 99 " 1 \+ 4 6 4 1 " }}{PARA 0 "" 0 "" {TEXT -1 106 " \+ 1 5 \+ 10 10 5 1 " }}{PARA 0 "" 0 "" {TEXT -1 114 " \+ 1 6 \+ 15 20 15 6 1 " }}{PARA 0 "" 0 "" {TEXT -1 123 " 1 \+ 7 21 35 35 21 7 1 " }}{PARA 0 "" 0 "" {TEXT -1 132 " \+ 1 8 28 56 70 56 28 \+ 8 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The sixth row gives the coefficients in the expansion of \+ " }{XPPEDIT 18 0 "(a+b)^6" "6#*$,&%\"aG\"\"\"%\"bGF&\"\"'" }{TEXT -1 12 ", which is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "( a+b)^6=a^6+6*a^5*b+15*a^4*b^2+20*a^3*b^3+15*a^2*b^4+6*a*b^5+b^6" "6#/* $,&%\"aG\"\"\"%\"bGF'\"\"',0*$F&F)F'*(F)F'*$F&\"\"&F'F(F'F'*(\"#:F'*$F &\"\"%F'F(\"\"#F'*(\"#?F'*$F&\"\"$F'F(F7F'*(F0F'*$F&F3F'F(F2F'*(F)F'F& F'F(F.F'*$F(F)F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((a+b)^6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"aG\"\"'\"\"\"F(*(F'F()F&\"\"&F(%\"bG F(F(*(\"#:F()F&\"\"%F()F,\"\"#F(F(*(\"#?F()F&\"\"$F()F,F6F(F(*(F.F()F& F2F()F,F0F(F(*(F'F(F&F()F,F+F(F(*$)F,F'F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Pascal's triangle can be constr ucted using Maple's built-in procedure " }{TEXT 0 8 "binomial" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "alias(C=binomial):\nn := 14:\nfor i from 1 to n do\n printf(StringTools[Join]([\"%5d \"$(i+1)]),seq(C(i,r),r=0..i));\n \+ printf(\"\\n\");\nend do;" }}{PARA 6 "" 1 "" {TEXT -1 15 " 1 \+ 1 " }}{PARA 6 "" 1 "" {TEXT -1 23 " 1 2 1 " }} {PARA 6 "" 1 "" {TEXT -1 31 " 1 3 3 1 " }}{PARA 6 "" 1 "" {TEXT -1 39 " 1 4 6 4 1 " }} {PARA 6 "" 1 "" {TEXT -1 47 " 1 5 10 10 5 \+ 1 " }}{PARA 6 "" 1 "" {TEXT -1 55 " 1 6 15 20 \+ 15 6 1 " }}{PARA 6 "" 1 "" {TEXT -1 63 " 1 7 21 35 35 21 7 1 " }}{PARA 6 "" 1 "" {TEXT -1 71 " 1 8 28 56 70 56 28 \+ 8 1 " }}{PARA 6 "" 1 "" {TEXT -1 79 " 1 9 36 \+ 84 126 126 84 36 9 1 " }}{PARA 6 " " 1 "" {TEXT -1 87 " 1 10 45 120 210 252 \+ 210 120 45 10 1 " }}{PARA 6 "" 1 "" {TEXT -1 95 " 1 11 55 165 330 462 462 330 165 \+ 55 11 1 " }}{PARA 6 "" 1 "" {TEXT -1 103 " 1 \+ 12 66 220 495 792 924 792 495 220 \+ 66 12 1 " }}{PARA 6 "" 1 "" {TEXT -1 111 " 1 13 \+ 78 286 715 1287 1716 1716 1287 715 28 6 78 13 1 " }}{PARA 6 "" 1 "" {TEXT -1 119 " 1 \+ 14 91 364 1001 2002 3003 3432 3003 2002 \+ 1001 364 91 14 1 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Binomial coefficients and the general binomial \+ expansion " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 37 "The coefficients in the expansion of " }{XPPEDIT 18 0 "(a+b)^n;" "6#),&%\"aG\"\"\"%\"bGF&%\"nG" }{TEXT -1 22 " are give n are called " }{TEXT 259 21 "binomial coefficients" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 32 "In general, the coefficient of " } {XPPEDIT 18 0 "a^i*b^j" "6#*&)%\"aG%\"iG\"\"\")%\"bG%\"jGF'" }{TEXT -1 22 " in the expansion of " }{XPPEDIT 18 0 "(a+b)^n" "6#),&%\"aG\" \"\"%\"bGF&%\"nG" }{TEXT -1 5 " is " }{TEXT 259 39 "the number of way s of selecting either " }{TEXT 288 1 "i" }{TEXT 259 12 " items from " }{TEXT 289 1 "n" }{TEXT -1 18 " or equivalently, " }{TEXT 278 1 "j" } {TEXT -1 12 " items from " }{TEXT 279 1 "n" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Hence this coefficient is " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "C[n,i] = C[n,j];" "6#/&%\"CG6$%\"nG% \"iG&F%6$F'%\"jG" }{XPPEDIT 18 0 "``=n!/(i!*j!)" "6#/%!G*&-%*factorial G6#%\"nG\"\"\"*&-F'6#%\"iGF*-F'6#%\"jGF*!\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "n! = n*(n-1)*` . . . `*3*`.`*2*`.`*1;" "6#/-%*factorialG6#%\"nG*2F'\"\"\",&F'F)F)!\"\"F) %(~.~.~.~GF)\"\"$F)%\".GF)\"\"#F)F.F)F)F)" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "A quick reminder \+ of where this comes from may be appropriate. If you do not need this r eminder then skip the next few lines. " }}{PARA 0 "" 0 "" {TEXT -1 22 "The number of ways of " }{TEXT 259 9 "arranging" }{TEXT -1 1 " " } {TEXT 294 1 "r" }{TEXT -1 20 " objects taken from " }{TEXT 295 1 "n" } {TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " n*(n-1)*(n-2)*` . . . `*(n-r+2)*(n-r+1) = n!/(n-r)!;" "6#/*.%\"nG\"\" \",&F%F&F&!\"\"F&,&F%F&\"\"#F(F&%(~.~.~.~GF&,(F%F&%\"rGF(F*F&F&,(F%F&F -F(F&F&F&*&-%*factorialG6#F%F&-F16#,&F%F&F-F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 22 "The number of ways of " }{TEXT 259 9 "sel ecting" }{TEXT -1 1 " " }{TEXT 296 1 "r" }{TEXT -1 14 " objects from \+ " }{TEXT 297 1 "n" }{TEXT -1 30 ", without regard to order, is " } {XPPEDIT 18 0 "n!/(n-r)!;" "6#*&-%*factorialG6#%\"nG\"\"\"-F%6#,&F'F(% \"rG!\"\"F-" }{TEXT -1 44 " divided by the number of ways of arranging " }{TEXT 298 1 "r" }{TEXT -1 49 " objects in a selection among themse lves, namely " }{XPPEDIT 18 0 "r!;" "6#-%*factorialG6#%\"rG" }{TEXT -1 13 ". This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C[n,r] = n!/(r!*(n-r)!);" "6#/&%\"CG6$%\"nG%\"rG*&-%*factorialG6#F' \"\"\"*&-F+6#F(F--F+6#,&F'F-F(!\"\"F-F4" }{XPPEDIT 18 0 "``=n*(n-1)*` \+ . . . `*(n-r+2)*(n-r+1)/r!" "6#/%!G*.%\"nG\"\"\",&F&F'F'!\"\"F'%(~.~.~ .~GF',(F&F'%\"rGF)\"\"#F'F',(F&F'F,F)F'F'F'-%*factorialG6#F,F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "To see why the coefficient of " }{XPPEDIT 18 0 "a^i*b^j " "6#*&)%\"aG%\"iG\"\"\")%\"bG%\"jGF'" }{TEXT -1 21 " in the expansion of " }{XPPEDIT 18 0 "(a+b)^n" "6#),&%\"aG\"\"\"%\"bGF&%\"nG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "C[n,j];" "6#&%\"CG6$%\"nG%\"jG" }{TEXT -1 28 ", consider the expansion of " }{XPPEDIT 18 0 "(a+b)^5;" "6#*$,&%\" aG\"\"\"%\"bGF&\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 " In expanding " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "(a+ b)^5 = (a+b)*(a+b)*(a+b)*(a+b)*(a+b);" "6#/*$,&%\"aG\"\"\"%\"bGF'\"\"& *,,&F&F'F(F'F',&F&F'F(F'F',&F&F'F(F'F',&F&F'F(F'F',&F&F'F(F'F'" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "we can obtain a term in " }{XPPEDIT 18 0 "a^4*b;" "6#*&%\"aG\"\"%%\"bG\"\"\"" }{TEXT -1 4 " i n " }{TEXT 259 6 "5 ways" }{TEXT -1 13 " as follows: " }}{PARA 15 "" 0 "" {TEXT -1 10 "Multiply \"" }{TEXT 280 1 "a" }{TEXT -1 52 "\"s from each of the 1st 4 brackets together with a \"" }{TEXT 281 1 "b" } {TEXT -1 33 "\" from the 5th bracket to obtain " }{XPPEDIT 18 0 "a*a*a *a*b=a^4*b" "6#/*,%\"aG\"\"\"F%F&F%F&F%F&%\"bGF&*&F%\"\"%F'F&" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 8 "Take a \"" }{TEXT 282 1 "b" } {TEXT -1 40 "\" from the 4th bracket multiplied with \"" }{TEXT 283 1 "a" }{TEXT -1 39 "\"s from the other 4 brackets to obtain " }{XPPEDIT 18 0 "a*a*a*b*a=a^4*b" "6#/*,%\"aG\"\"\"F%F&F%F&%\"bGF&F%F&*&F%\"\"%F' F&" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 8 "Take a \"" }{TEXT 284 1 "b" }{TEXT -1 40 "\" from the 3rd bracket multiplied with \"" } {TEXT 285 1 "a" }{TEXT -1 39 "\"s from the other 4 brackets to obtain \+ " }{XPPEDIT 18 0 "a*a*b*a*a=a^4*b" "6#/*,%\"aG\"\"\"F%F&%\"bGF&F%F&F%F &*&F%\"\"%F'F&" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 8 "Take a \"" }{TEXT 292 1 "b" }{TEXT -1 40 "\" from the 2nd bracket multiplied with \"" }{TEXT 290 1 "a" }{TEXT -1 39 "\"s from the other 4 brackets to obtain " }{XPPEDIT 18 0 "a*b*a*a*a=a^4*b" "6#/*,%\"aG\"\"\"%\"bGF& F%F&F%F&F%F&*&F%\"\"%F'F&" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 8 "Take a \"" }{TEXT 293 1 "b" }{TEXT -1 40 "\" from the 1st bracke t multiplied with \"" }{TEXT 291 1 "a" }{TEXT -1 39 "\"s from the othe r 4 brackets to obtain " }{XPPEDIT 18 0 "b*a*a*a*a=a^4*b" "6#/*,%\"bG \"\"\"%\"aGF&F'F&F'F&F'F&*&F'\"\"%F%F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of \+ " }{XPPEDIT 18 0 "a^4*b;" "6#*&%\"aG\"\"%%\"bG\"\"\"" }{TEXT -1 101 " \+ is the number of ways of selecting one bracket from the 5 brackets, f rom which to take the single \"" }{TEXT 286 1 "b" }{TEXT -1 2 "\"." }} {PARA 0 "" 0 "" {TEXT -1 110 "Equivalently, it is the number of ways o f selecting 4 brackets from the 5 brackets, from which to take the 4 \+ \"" }{TEXT 287 1 "a" }{TEXT -1 3 "\"s." }}{PARA 0 "" 0 "" {TEXT -1 16 "Since there are " }{XPPEDIT 18 0 "C[5,1] = 5;" "6#/&%\"CG6$\"\"&\"\" \"F'" }{TEXT -1 19 " ways of obtaining " }{XPPEDIT 18 0 "a^4*b;" "6#*& %\"aG\"\"%%\"bG\"\"\"" }{TEXT -1 45 ", the corresponding term in the e xpansion of " }{XPPEDIT 18 0 "(a+b)^5" "6#*$,&%\"aG\"\"\"%\"bGF&\"\"& " }{TEXT -1 4 " is " }{XPPEDIT 18 0 "5*a^4*b;" "6#*(\"\"&\"\"\"*$%\"aG \"\"%F%%\"bGF%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 40 "For another example, the coefficient of \+ " }{XPPEDIT 18 0 "a^3*b^2;" "6#*&%\"aG\"\"$%\"bG\"\"#" }{TEXT -1 21 " \+ in the expansion of " }{XPPEDIT 18 0 "(a+b)^5;" "6#*$,&%\"aG\"\"\"%\"b GF&\"\"&" }{TEXT -1 98 " is the number of ways of selecting two bracke ts from the 5 brackets, from which to take the two \"" }{TEXT 299 1 "b " }{TEXT -1 39 "\"s. This means that the coefficient of " }{XPPEDIT 18 0 "a^3*b^2;" "6#*&%\"aG\"\"$%\"bG\"\"#" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "C[5,2] = 5!/(2!*3!);" "6#/&%\"CG6$\"\"&\"\"#*&-%*factor ialG6#F'\"\"\"*&-F+6#F(F--F+6#\"\"$F-!\"\"" }{XPPEDIT 18 0 "``=5*`.`*4 /(2*`.`*1)" "6#/%!G**\"\"&\"\"\"%\".GF'\"\"%F'*(\"\"#F'F(F'F'F'!\"\"" }{XPPEDIT 18 0 "``=10" "6#/%!G\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "We may associate the 10 selections with the following 10 products. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a*a*a*b*b" "6#*,%\"aG\"\"\"F$F%F$F%%\"b GF%F&F%" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a*a*b*a*b" "6#*,%\"aG\"\" \"F$F%%\"bGF%F$F%F&F%" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a*b*a*a*b" " 6#*,%\"aG\"\"\"%\"bGF%F$F%F$F%F&F%" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "b*a*a*a*b" "6#*,%\"bG\"\"\"%\"aGF%F&F%F&F%F$F%" }{TEXT -1 5 ", " } {XPPEDIT 18 0 "a*a*b*b*a" "6#*,%\"aG\"\"\"F$F%%\"bGF%F&F%F$F%" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a*b*a*b*a" "6#*,%\"aG\"\"\"%\"bGF%F$F%F&F% F$F%" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "b*a*a*b*a" "6#*,%\"bG\"\"\"% \"aGF%F&F%F$F%F&F%" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "a*b*b*a*a" "6#* ,%\"aG\"\"\"%\"bGF%F&F%F$F%F$F%" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "b* a*b*a*a" "6#*,%\"bG\"\"\"%\"aGF%F$F%F&F%F&F%" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "b*b*a*a*a" "6#*,%\"bG\"\"\"F$F%%\"aGF%F&F%F&F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^ 5 = Sum(C[5,j]*a^(5-j)*b^j,j = 0 .. 5);" "6#/*$,&%\"aG\"\"\"%\"bGF'\" \"&-%$SumG6$*(&%\"CG6$F)%\"jGF')F&,&F)F'F1!\"\"F')F(F1F'/F1;\"\"!F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = C[5,0]*a^5+C[5,1]*a^4*b+C[5,2]*a^3 *b^2+C[5,3]*a^2*b^3+C[5,4]*a*b^4+C[5,5]*b^5;" "6#/%!G,.*&&%\"CG6$\"\"& \"\"!\"\"\"*$%\"aGF*F,F,*(&F(6$F*F,F,*$F.\"\"%F,%\"bGF,F,*(&F(6$F*\"\" #F,*$F.\"\"$F,F4F8F,*(&F(6$F*F:F,*$F.F8F,F4F:F,*(&F(6$F*F3F,F.F,F4F3F, *&&F(6$F*F*F,*$F4F*F,F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=a^5+5*a^4*b+1 0*a^3*b^2+10*a^2*b^3+5*a*b^4+b^5" "6#/%!G,.*$%\"aG\"\"&\"\"\"*(F(F)*$F '\"\"%F)%\"bGF)F)*(\"#5F)*$F'\"\"$F)F-\"\"#F)*(F/F)*$F'F2F)F-F1F)*(F(F )F'F)F-F,F)*$F-F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The notation " }{XPPEDIT 18 0 "C[n,j] = matrix([[n], [j]]);" "6#/&%\"CG6$%\"nG%\"jG-%'matrixG6#7$7#F'7#F(" } {XPPEDIT 18 0 "``=n!/(j!*(n-j)!)" "6#/%!G*&-%*factorialG6#%\"nG\"\"\"* &-F'6#%\"jGF*-F'6#,&F)F*F.!\"\"F*F2" }{TEXT -1 60 " is generally used \+ in conjunction with binomial expansions. " }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^5=S um(matrix([[5],[j]])*a^(5-j)*b^j,j=0..5)" "6#/*$,&%\"aG\"\"\"%\"bGF'\" \"&-%$SumG6$*(-%'matrixG6#7$7#F)7#%\"jGF')F&,&F)F'F4!\"\"F')F(F4F'/F4; \"\"!F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=matrix([[5], [0]])*a^5+matrix ([[5], [1]])*a^4*b+matrix([[5], [2]])*a^3*b^2+matrix([[5], [3]])*a^2*b ^3+matrix([[5], [4]])*a*b^4+matrix([[5], [5]])*b^5" "6#/%!G,.*&-%'matr ixG6#7$7#\"\"&7#\"\"!\"\"\"*$%\"aGF,F/F/*(-F(6#7$7#F,7#F/F/*$F1\"\"%F/ %\"bGF/F/*(-F(6#7$7#F,7#\"\"#F/*$F1\"\"$F/F:FAF/*(-F(6#7$7#F,7#FCF/*$F 1FAF/F:FCF/*(-F(6#7$7#F,7#F9F/F1F/F:F9F/*&-F(6#7$7#F,7#F,F/*$F:F,F/F/ " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "In general, the coefficient of " }{XPPEDIT 18 0 "a^(n-j) *b^j" "6#*&)%\"aG,&%\"nG\"\"\"%\"jG!\"\"F()%\"bGF)F(" }{TEXT -1 21 " i n the expansion of " }{XPPEDIT 18 0 "(a+b)^n" "6#),&%\"aG\"\"\"%\"bGF& %\"nG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "C[n,j] = matrix([[n], [j]]); " "6#/&%\"CG6$%\"nG%\"jG-%'matrixG6#7$7#F'7#F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 7 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(a+b)^n = Sum(matrix([[n], [j]])*a^(n-j)*b^j,j = 0 . . n)" "6#/),&%\"aG\"\"\"%\"bGF'%\"nG-%$SumG6$*(-%'matrixG6#7$7#F)7#%\" jGF')F&,&F)F'F4!\"\"F')F(F4F'/F4;\"\"!F)" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = matrix([[n], [0]])*a^n+matrix([[n], [1]])*a^(n-1)*b+matrix([[n ], [2]])*a^(n-2)*b^2+` . . . `+matrix([[n], [n-1]])*a*b^(n-1)+matrix([ [n], [n]])*b^n;" "6#/%!G,.*&-%'matrixG6#7$7#%\"nG7#\"\"!\"\"\")%\"aGF, F/F/*(-F(6#7$7#F,7#F/F/)F1,&F,F/F/!\"\"F/%\"bGF/F/*(-F(6#7$7#F,7#\"\"# F/)F1,&F,F/FBF:F/F;FBF/%(~.~.~.~GF/*(-F(6#7$7#F,7#,&F,F/F/F:F/F1F/)F;, &F,F/F/F:F/F/*&-F(6#7$7#F,7#F,F/)F;F,F/F/" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 300 48 "____________________________ ____________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "wh ere " }{XPPEDIT 18 0 "matrix([[n], [j]])=n!/(j!*(n-j)!)" "6#/-%'matrix G6#7$7#%\"nG7#%\"jG*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0-F.6#,&F)F0F+! \"\"F0F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "matrix([[n], [j]]) = matrix([[n], [n-j]])" "6#/-%'matr ixG6#7$7#%\"nG7#%\"jG-F%6#7$7#F)7#,&F)\"\"\"F+!\"\"" }{TEXT -1 7 " fo r " }{XPPEDIT 18 0 "j=0,1,` . . . `,n" "6&/%\"jG\"\"!\"\"\"%(~.~.~.~G %\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "For example, the coefficient of " }{XPPEDIT 18 0 "a^ 5*b^3;" "6#*&%\"aG\"\"&%\"bG\"\"$" }{TEXT -1 21 " in the expansion of \+ " }{XPPEDIT 18 0 "(a+b)^8;" "6#*$,&%\"aG\"\"\"%\"bGF&\"\")" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "C[8,3]=matrix([[8],[3])" "6#/&%\"CG6$\"\")\" \"$-%'matrixG6#7$7#F'7#F(" }{XPPEDIT 18 0 "``=8!/(3!*`.`*5!)" "6#/%!G* &-%*factorialG6#\"\")\"\"\"*(-F'6#\"\"$F*%\".GF*-F'6#\"\"&F*!\"\"" } {XPPEDIT 18 0 "``=8*`.`*7*`.`*6/(3*`.`*2*`.`*1)" "6#/%!G*.\"\")\"\"\"% \".GF'\"\"(F'F(F'\"\"'F'*,\"\"$F'F(F'\"\"#F'F(F'F'F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``= 56" "6#/%!G\"#c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "a^4*b^4;" "6 #*&%\"aG\"\"%%\"bGF%" }{TEXT -1 21 " in the expansion of " }{XPPEDIT 18 0 "(a+b)^8;" "6#*$,&%\"aG\"\"\"%\"bGF&\"\")" }{TEXT -1 4 " is " } {XPPEDIT 18 0 "C[8,4] = matrix([[8], [4]]);" "6#/&%\"CG6$\"\")\"\"%-%' matrixG6#7$7#F'7#F(" }{XPPEDIT 18 0 "`` = 8!/(4!*`.`*4!);" "6#/%!G*&-% *factorialG6#\"\")\"\"\"*(-F'6#\"\"%F*%\".GF*-F'6#F.F*!\"\"" } {XPPEDIT 18 0 "`` = 8*`.`*7*`.`*6*`.`*5/(4*`.`*3*`.`*2*`.`*1);" "6#/%! G*2\"\")\"\"\"%\".GF'\"\"(F'F(F'\"\"'F'F(F'\"\"&F'*0\"\"%F'F(F'\"\"$F' F(F'\"\"#F'F(F'F'F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 70;" "6 #/%!G\"#q" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "The coeffic ient of " }{XPPEDIT 18 0 "a^3*b^5;" "6#*&%\"aG\"\"$%\"bG\"\"&" }{TEXT -1 21 " in the expansion of " }{XPPEDIT 18 0 "(a+b)^8;" "6#*$,&%\"aG\" \"\"%\"bGF&\"\")" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "matrix([[8], [5]] )=matrix([[8], [3]])" "6#/-%'matrixG6#7$7#\"\")7#\"\"&-F%6#7$7#F)7#\" \"$" }{XPPEDIT 18 0 "``=56" "6#/%!G\"#c" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The complete expan sion of " }{XPPEDIT 18 0 "(a+b)^8;" "6#*$,&%\"aG\"\"\"%\"bGF&\"\")" } {TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " (a+b)^8=a^8+8*a^7*b+28*a^6*b^2+56*a^5*b^3+70*a^4*b^4+56*a^3*b^5+28*a^2 *b^6+8*a*b^7+b^8" "6#/*$,&%\"aG\"\"\"%\"bGF'\"\"),4*$F&F)F'*(F)F'*$F& \"\"(F'F(F'F'*(\"#GF'*$F&\"\"'F'F(\"\"#F'*(\"#cF'*$F&\"\"&F'F(\"\"$F'* (\"#qF'*$F&\"\"%F'F(F " 0 "" {MPLTEXT 1 0 31 "sum(C[5,j]*a^(5-j)*b^j,j=0.. 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&&%\"CG6$\"\"&\"\"!\"\"\")% \"aGF(F*F**(&F&6$F(F*F*)F,\"\"%F*%\"bGF*F**(&F&6$F(\"\"#F*)F,\"\"$F*)F 2F6F*F**(&F&6$F(F8F*)F,F6F*)F2F8F*F**(&F&6$F(F1F*F,F*)F2F1F*F**&&F&6$F (F(F*)F2F(F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "sum(matrix([[5],[j]])*a^(5-j)*b^j,j=0..5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&K%'matrixG6#7$7#\"\"&7#\"\"!Q(ppri nt06\"\"\"\")%\"aGF*F/F/*(KF&6#7$F)7#F/Q(pprint1F.F/)F1\"\"%F/%\"bGF/F /*(KF&6#7$F)7#\"\"#Q(pprint2F.F/)F1\"\"$F/)F:F@F/F/*(KF&6#7$F)7#FCQ(pp rint3F.F/)F1F@F/)F:FCF/F/*(KF&6#7$F)7#F9Q(pprint4F.F/F1F/)F:F9F/F/*&KF &6#7$F)F)Q(pprint5F.F/)F:F*F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "n := 8:\nr := 3:\nn!/(r!*(n- r)!);\nbinomial(n,r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#c" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#c" }}}{PARA 0 "" 0 "" {TEXT -1 26 " The complete expansion of " }{XPPEDIT 18 0 "(a+b)^8;" "6#*$,&%\"aG\"\" \"%\"bGF&\"\")" }{TEXT -1 4 " is:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "(a+b)^8;\n``=expand(%); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&%\"aG\"\"\"%\"bGF'\"\")F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,4*$)%\"aG\"\")\"\"\"F**(F)F*)F(\" \"(F*%\"bGF*F**(\"#GF*)F(\"\"'F*)F.\"\"#F*F**(\"#cF*)F(\"\"&F*)F.\"\"$ F*F**(\"#qF*)F(\"\"%F*)F.F>F*F**(F6F*)F(F:F*)F.F8F*F**(F0F*)F(F4F*)F.F 2F*F**(F)F*F(F*)F.F-F*F**$)F.F)F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 304 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 23 "Find the expansion of " }{XPPEDIT 18 0 "(x/2-y)^7;" "6#*$,&*&% \"xG\"\"\"\"\"#!\"\"F'%\"yGF)\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 305 8 "Solution" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 31 "First we find the expansion of " } {XPPEDIT 18 0 "(a+b)^7;" "6#*$,&%\"aG\"\"\"%\"bGF&\"\"(" }{TEXT -1 2 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^7 = Sum(m atrix([[7], [j]])*a^(7-j)*b^j,j = 0 .. 7);" "6#/*$,&%\"aG\"\"\"%\"bGF' \"\"(-%$SumG6$*(-%'matrixG6#7$7#F)7#%\"jGF')F&,&F)F'F4!\"\"F')F(F4F'/F 4;\"\"!F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=matrix([[7], [0]])*a^7+m atrix([[7], [1]])*a^6*b+matrix([[7], [2]])*a^5*b^2+matrix([[7], [3]])* a^4*b^3+matrix([[7], [4]])*a^3*b^4+matrix([[7], [5]])*a^2*b^5+matrix([ [7], [6]])*a*b^6+matrix([[7], [7]])*b^7" "6#/%!G,2*&-%'matrixG6#7$7#\" \"(7#\"\"!\"\"\"*$%\"aGF,F/F/*(-F(6#7$7#F,7#F/F/*$F1\"\"'F/%\"bGF/F/*( -F(6#7$7#F,7#\"\"#F/*$F1\"\"&F/F:FAF/*(-F(6#7$7#F,7#\"\"$F/*$F1\"\"%F/ F:FJF/*(-F(6#7$7#F,7#FLF/*$F1FJF/F:FLF/*(-F(6#7$7#F,7#FCF/*$F1FAF/F:FC F/*(-F(6#7$7#F,7#F9F/F1F/F:F9F/*&-F(6#7$7#F,7#F,F/*$F:F,F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "The binomial coefficients can be evaluated individually by means o f the formula e " }{XPPEDIT 18 0 "matrix([[n], [j]])=n!/(j!*(n-j)!)" " 6#/-%'matrixG6#7$7#%\"nG7#%\"jG*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0-F. 6#,&F)F0F+!\"\"F0F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "F or example, " }{XPPEDIT 18 0 "matrix([[7], [3]]) = 7!/(3!*`.`*4!);" "6 #/-%'matrixG6#7$7#\"\"(7#\"\"$*&-%*factorialG6#F)\"\"\"*(-F.6#F+F0%\". GF0-F.6#\"\"%F0!\"\"" }{XPPEDIT 18 0 "`` = 7*`.`*6*`.`*5/(3*`.`*2*`.`* 1);" "6#/%!G*.\"\"(\"\"\"%\".GF'\"\"'F'F(F'\"\"&F'*,\"\"$F'F(F'\"\"#F' F(F'F'F'!\"\"" }{XPPEDIT 18 0 "`` = 35;" "6#/%!G\"#N" }{TEXT -1 3 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 86 "However, it is probably simpler to co nstruct Pascal's triangle as far as the 7th row. " }}{PARA 0 "" 0 "" {TEXT -1 37 "The coefficients in the expansion of " }{XPPEDIT 18 0 "(a +b)^7;" "6#*$,&%\"aG\"\"\"%\"bGF&\"\"(" }{TEXT -1 6 " are: " }}{PARA 256 "" 0 "" {TEXT -1 66 " 1 7 21 35 35 \+ 21 7 1 " }}{PARA 0 "" 0 "" {TEXT -1 26 "This gives the e xpansion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^7 =a^7+7*a^6*b+21*a^5*b^2+35*a^4*b^3+35*a^3*b^4+21*a^2*b^5+7*a*b^6+b^7" "6#/*$,&%\"aG\"\"\"%\"bGF'\"\"(,2*$F&F)F'*(F)F'*$F&\"\"'F'F(F'F'*(\"#@ F'*$F&\"\"&F'F(\"\"#F'*(\"#NF'*$F&\"\"%F'F(\"\"$F'*(F5F'*$F&F8F'F(F7F' *(F0F'*$F&F3F'F(F2F'*(F)F'F&F'F(F.F'*$F(F)F'" }{TEXT -1 2 ". " }{TEXT 306 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "To obtain the expansion of " }{XPPEDIT 18 0 "(x/2-y )^7;" "6#*$,&*&%\"xG\"\"\"\"\"#!\"\"F'%\"yGF)\"\"(" }{TEXT -1 23 " we \+ need to substitute " }{XPPEDIT 18 0 "a = x/2;" "6#/%\"aG*&%\"xG\"\"\" \"\"#!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = -y;" "6#/%\"bG,$% \"yG!\"\"" }{TEXT -1 28 " in the previous expansion. " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "(x/2-y)^7 = (x/2)^7+7*(x/2)^6*(-y)+21*(x/2)^5*(-y)^2+35 *(x/2)^4*(-y)^3+35*(x/2)^3*(-y)^4+21*(x/2)^2*(-y)^5+7*``(x/2)*(-y)^6+( -y)^7;" "6#/*$,&*&%\"xG\"\"\"\"\"#!\"\"F(%\"yGF*\"\"(,2*$*&F'F(F)F*F,F (*(F,F(*$*&F'F(F)F*\"\"'F(,$F+F*F(F(*(\"#@F(*$*&F'F(F)F*\"\"&F(,$F+F*F )F(*(\"#NF(*$*&F'F(F)F*\"\"%F(,$F+F*\"\"$F(*(F " 0 "" {MPLTEXT 1 0 26 "seq(binomial(7,j),j=0..7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*\"\"\"\"\"(\"#@\"#NF&F%F$F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(a +b)^7;\n``=expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&%\"aG\" \"\"%\"bGF'\"\"(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,2*$)%\"aG\" \"(\"\"\"F**(F)F*)F(\"\"'F*%\"bGF*F**(\"#@F*)F(\"\"&F*)F.\"\"#F*F**(\" #NF*)F(\"\"%F*)F.\"\"$F*F**(F6F*)F(F:F*)F.F8F*F**(F0F*)F(F4F*)F.F2F*F* *(F)F*F(F*)F.F-F*F**$)F.F)F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(x/2-y)^7;\n``=expand(%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&*&\"\"#!\"\"%\"xG\"\"\"F*%\"yGF( \"\"(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,2*&#\"\"\"\"$G\"F(*$)% \"xG\"\"(F(F(F(*&#F-\"#kF(*&)F,\"\"'F(%\"yGF(F(!\"\"*&#\"#@\"#KF(*&)F, \"\"&F()F4\"\"#F(F(F(*&#\"#N\"#;F(*&)F,\"\"%F()F4\"\"$F(F(F5*&#FA\"\") F(*&)F,FGF()F4FEF(F(F(*&#F8FEF(*&)F,F>F()F4FF(*&F,F()F4F 3F(F(F(*$)F4F-F(F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 " Example 2 " }}{PARA 0 "" 0 "" {TEXT 301 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 22 "Find the expansion of " }{XPPEDIT 18 0 "(x-2*y)^10" "6#*$,&%\"xG\"\"\"*&\"\"#F&%\"yGF&!\"\"\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 8 "So lution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 "First we find \+ the expansion of " }{XPPEDIT 18 0 "(a+b)^10" "6#*$,&%\"aG\"\"\"%\"bGF& \"#5" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^10 = Sum(matrix([[10], [j]])*a^(10-j)*b^j,j = 0 .. 10);" " 6#/*$,&%\"aG\"\"\"%\"bGF'\"#5-%$SumG6$*(-%'matrixG6#7$7#F)7#%\"jGF')F& ,&F)F'F4!\"\"F')F(F4F'/F4;\"\"!F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=ma trix([[10], [0]])*a^10+matrix([[10], [1]])*a^9*b+matrix([[10], [2]])*a ^8*b^2+` . . . `+matrix([[10], [9]])*a*b^9+matrix([[10], [10]])*b^10" "6#/%!G,.*&-%'matrixG6#7$7#\"#57#\"\"!\"\"\"*$%\"aGF,F/F/*(-F(6#7$7#F, 7#F/F/*$F1\"\"*F/%\"bGF/F/*(-F(6#7$7#F,7#\"\"#F/*$F1\"\")F/F:FAF/%(~.~ .~.~GF/*(-F(6#7$7#F,7#F9F/F1F/F:F9F/*&-F(6#7$7#F,7#F,F/*$F:F,F/F/" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Since " }{XPPEDIT 18 0 "matrix([[10], [j]]) = matrix([[10 ], [10-j]]);" "6#/-%'matrixG6#7$7#\"#57#%\"jG-F%6#7$7#F)7#,&F)\"\"\"F+ !\"\"" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "j = 0,1,` . . . `,10;" "6 &/%\"jG\"\"!\"\"\"%(~.~.~.~G\"#5" }{TEXT -1 51 ". , we only need to fi nd the binomial coefficients " }{XPPEDIT 18 0 "matrix([[10], [j]])" "6 #-%'matrixG6#7$7#\"#57#%\"jG" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "j=0 ,1,2,3,4,5" "6(/%\"jG\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[10], [0]])=1" "6#/-%'matrixG6#7$7#\"#57#\"\"! \"\"\"" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "matrix([[10], [1]])=10!/(1 !*9!)" "6#/-%'matrixG6#7$7#\"#57#\"\"\"*&-%*factorialG6#F)F+*&-F.6#F+F +-F.6#\"\"*F+!\"\"" }{XPPEDIT 18 0 "``=10" "6#/%!G\"#5" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "matrix([[10], [2]])=10!/(2!*8!)" "6#/-%'matrixG6 #7$7#\"#57#\"\"#*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0-F.6#\"\")F0!\"\" " }{XPPEDIT 18 0 "``=10*`.`*9/(2*`.`*1)" "6#/%!G**\"#5\"\"\"%\".GF'\" \"*F'*(\"\"#F'F(F'F'F'!\"\"" }{XPPEDIT 18 0 "``=45" "6#/%!G\"#X" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "matrix([[10], [3]])=10!/(3!*7!)" "6 #/-%'matrixG6#7$7#\"#57#\"\"$*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0-F.6# \"\"(F0!\"\"" }{XPPEDIT 18 0 "``=10*`.`*9*`.`*8/(3*`.`*2*`.`*1)" "6#/% !G*.\"#5\"\"\"%\".GF'\"\"*F'F(F'\"\")F'*,\"\"$F'F(F'\"\"#F'F(F'F'F'!\" \"" }{XPPEDIT 18 0 "``=120" "6#/%!G\"$?\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[10], [4]])=10!/(4!*6!)" "6#/-%'matrixG6#7$7#\"#57#\"\"%*&-%* factorialG6#F)\"\"\"*&-F.6#F+F0-F.6#\"\"'F0!\"\"" }{XPPEDIT 18 0 "`` = 10*`.`*9*`.`*8*`.`*7/(4*`.`*3*`.`*2*`.`*1)" "6#/%!G*2\"#5\"\"\"%\".GF '\"\"*F'F(F'\"\")F'F(F'\"\"(F'*0\"\"%F'F(F'\"\"$F'F(F'\"\"#F'F(F'F'F'! \"\"" }{XPPEDIT 18 0 "``=10*`.`*3*`.`*7" "6#/%!G*,\"#5\"\"\"%\".GF'\" \"$F'F(F'\"\"(F'" }{XPPEDIT 18 0 "``=210" "6#/%!G\"$5#" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "matrix([[10], [5]]) = 10!/(5!*5!);" "6#/-%'matr ixG6#7$7#\"#57#\"\"&*&-%*factorialG6#F)\"\"\"*&-F.6#F+F0-F.6#F+F0!\"\" " }{XPPEDIT 18 0 "`` = 10*`.`*9*`.`*8*`.`*7*`.`*6/(5*`.`*4*`.`*3*`.`*2 *`.`*1)" "6#/%!G*6\"#5\"\"\"%\".GF'\"\"*F'F(F'\"\")F'F(F'\"\"(F'F(F'\" \"'F'*4\"\"&F'F(F'\"\"%F'F(F'\"\"$F'F(F'\"\"#F'F(F'F'F'!\"\"" } {XPPEDIT 18 0 "``=2*`.`*3*`.`*7*`.`*6" "6#/%!G*0\"\"#\"\"\"%\".GF'\"\" $F'F(F'\"\"(F'F(F'\"\"'F'" }{XPPEDIT 18 0 "``=252" "6#/%!G\"$_#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "matrix([[10], [6]]) =matrix([[10], \+ [4]])" "6#/-%'matrixG6#7$7#\"#57#\"\"'-F%6#7$7#F)7#\"\"%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=210" "6#/%!G\"$5#" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "matrix([[10], [7]]) =matrix([[10], [3]]) " "6#/-%'matri xG6#7$7#\"#57#\"\"(-F%6#7$7#F)7#\"\"$" }{XPPEDIT 18 0 "``=120" "6#/%!G \"$?\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "matrix([[10], [8]]) =matrix ([[10], [2]]) " "6#/-%'matrixG6#7$7#\"#57#\"\")-F%6#7$7#F)7#\"\"#" } {XPPEDIT 18 0 "``=45" "6#/%!G\"#X" }{TEXT -1 4 ", " }{XPPEDIT 18 0 " matrix([[10], [9]]) =matrix([[10], [1]])" "6#/-%'matrixG6#7$7#\"#57#\" \"*-F%6#7$7#F)7#\"\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=10" "6#/%!G \"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "matrix([[10], [10]]) =1" " 6#/-%'matrixG6#7$7#\"#57#F)\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Summarising, the coeffici ents in the expansion of " }{XPPEDIT 18 0 "(a+b)^10" "6#*$,&%\"aG\"\" \"%\"bGF&\"#5" }{TEXT -1 6 " are: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 80 "1 10 45 120 \+ 210 252 210 " }{TEXT 303 0 "" }{TEXT -1 37 "120 45 10 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 106 "These binomial coefficients could al so be obtained by extending Pascal's triangle as far as the 10th row. \+ " }}{PARA 0 "" 0 "" {TEXT -1 26 "This gives the expansion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a+b)^10 = a^10+10*a^9*b+45 *a^8*b^2+120*a^7*b^3+210*a^6*b^4+252*a^5*b^5+210*a^4*b^6+120*a^3*b^7+4 5*a^2*b^8+10*a*b^9+b^10;" "6#/*$,&%\"aG\"\"\"%\"bGF'\"#5,8*$F&F)F'*(F) F'*$F&\"\"*F'F(F'F'*(\"#XF'*$F&\"\")F'F(\"\"#F'*(\"$?\"F'*$F&\"\"(F'F( \"\"$F'*(\"$5#F'*$F&\"\"'F'F(\"\"%F'*(\"$_#F'*$F&\"\"&F'F(FAF'*(F:F'*$ F&F=F'F(FF)F.F=F)*(\"&g`\"F)*$F'F9F)F.F8F/*(\"&?:\"F)*$F'F4F)F.F3F)*(\"%?^F)F' F)F.F-F/*&\"%C5F)*$F.F(F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "seq(binomial(10,j ),j=0..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"\"#5\"#X\"$?\"\"$ 5#\"$_#F'F&F%F$F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "(a+b)^10;\n``=expand(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*$),&%\"aG\"\"\"%\"bGF'\"#5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,8*$)%\"aG\"#5\"\"\"F(*(F'F()F&\"\"*F(%\"bGF(F(*(\"#XF( )F&\"\")F()F,\"\"#F(F(*(\"$?\"F()F&\"\"(F()F,\"\"$F(F(*(\"$5#F()F&\"\" 'F()F,\"\"%F(F(*(\"$_#F()F&\"\"&F()F,FBF(F(*(F:F()F&F>F()F,F " 0 "" {MPLTEXT 1 0 25 "(x-2*y)^10;\n``=expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&%\"xG\"\"\"*&\"\"#F'%\"yGF'!\"\"\"#5F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,8*$)%\"xG\"#5\"\"\"F**(\"#?F*)F(\"\"*F* %\"yGF*!\"\"*(\"$!=F*)F(\"\")F*)F/\"\"#F*F**(\"$g*F*)F(\"\"(F*)F/\"\"$ F*F0*(\"%gLF*)F(\"\"'F*)F/\"\"%F*F**(\"%k!)F*)F(\"\"&F*)F/FFF*F0*(\"&S M\"F*)F(FBF*)F/F@F*F**(\"&g`\"F*)F(F " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 41 "A relation between binomial coefficients " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Pascal's triangle .. 1" }}{PARA 0 "" 0 " " {TEXT -1 20 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "Le t's look more closely at the multiplication " }{XPPEDIT 18 0 "(a+b)*`. `*(a+b)^3 = (a+b)^4;" "6#/*(,&%\"aG\"\"\"%\"bGF'F'%\".GF',&F&F'F(F'\" \"$*$,&F&F'F(F'\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "expr1 := expand(a*(a+b)^3) ;\nexpr2 := expand(b*(a+b)^3);\nexpr1+expr2;" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 113 "It is easy to check th is last addition by arranging like terms of the two expressions in the same vertical line. " }}{PARA 0 "" 0 "" {TEXT -1 38 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 48 " \+ " }{XPPEDIT 18 0 "a^4+3*a^3*b+3*a^2*b ^2+a*b^3" "6#,**$%\"aG\"\"%\"\"\"*(\"\"$F'*$F%F)F'%\"bGF'F'*(F)F'*$F% \"\"#F'F+F.F'*&F%F'*$F+F)F'F'" }}{PARA 0 "" 0 "" {TEXT -1 59 " \+ " }{XPPEDIT 18 0 "a ^3*b+3*a^2*b^2+3*a*b^3+b^4;" "6#,**&%\"aG\"\"$%\"bG\"\"\"F(*(F&F(*$F% \"\"#F(F'F+F(*(F&F(F%F(F'F&F(*$F'\"\"%F(" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 122 " __ ____________________________ \+ " }}{PARA 0 "" 0 "" {TEXT -1 47 " \+ " }{XPPEDIT 18 0 "a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4;" "6# ,,*$%\"aG\"\"%\"\"\"*(F&F'*$F%\"\"$F'%\"bGF'F'*(\"\"'F'*$F%\"\"#F'F+F/ F'*(F&F'F%F'F+F*F'*$F+F&F'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Looking at the coefficients alone we have:" }} {PARA 0 "" 0 "" {TEXT -1 85 " \+ 1 3 3 1" }}{PARA 0 "" 0 "" {TEXT -1 96 " \+ 1 3 3 1 " }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ _________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 95 " \+ 1 4 6 4 1" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Instead \+ of writing the line of coefficients 1, 3, 3, 1 twice, we can generat e the new line of coefficients by using the addition scheme:" }}{PARA 0 "" 0 "" {TEXT -1 75 " \+ p -> q" }}{PARA 0 "" 0 "" {TEXT -1 77 " \+ |" }}{PARA 0 "" 0 "" {TEXT -1 78 " \+ p + q" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "app lied to the single row 1, 3, 3, 1. " }}{PARA 0 "" 0 "" {TEXT -1 68 "W e can obtain the coefficients in the expansion of in a similar way." } }{PARA 0 "" 0 "" {TEXT -1 102 " \+ 0 1 4 6 4 1 0" }} {PARA 0 "" 0 "" {TEXT -1 101 " \+ 1 5 10 10 5 1" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Zeros nee d to be inserted as shown at least mentally." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Starting with the coefficients \+ of1,1 of a + b and 1, 2, 1 of " }{XPPEDIT 18 0 "(a+b)^2 = a^2+2*a*b+b^ 2;" "6#/*$,&%\"aG\"\"\"%\"bGF'\"\"#,(*$F&F)F'*(F)F'F&F'F(F'F'*$F(F)F' " }{TEXT -1 17 " we can generate " }{TEXT 259 17 "Pascal's triangle" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 " 1 \+ 1 " }}{PARA 0 "" 0 "" {TEXT -1 74 " \+ 1 2 1 " }}{PARA 0 "" 0 "" {TEXT -1 83 " 1 \+ 3 3 1 " }}{PARA 0 "" 0 "" {TEXT -1 94 " \+ 1 4 6 \+ 4 1 " }}{PARA 0 "" 0 "" {TEXT -1 101 " \+ 1 5 10 10 5 \+ 1 " }}{PARA 0 "" 0 "" {TEXT -1 108 " \+ 1 6 15 20 15 \+ 6 1" }}{PARA 0 "" 0 "" {TEXT -1 117 " \+ 1 7 21 35 35 \+ 21 7 1" }}{PARA 0 "" 0 "" {TEXT -1 126 " \+ 1 8 28 56 \+ 70 56 28 8 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "expand((a+b)^8);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We can co nstruct Pascal's triangle using Maple's built-in procedure " }{TEXT 0 8 "binomial" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "alias(C=binomial):\nn := 18:\nfor i from 1 to n do\n print(seq(C(i,r),r=0..i));\nend do;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$\"\"\"F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\" \"\"\"#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&\"\"\"\"\"$F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'\"\"\"\"\"%\"\"'F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"\"\"\"&\"#5F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6)\"\"\"\"\"'\"#:\"#?F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*\"\"\" \"\"(\"#@\"#NF&F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+\"\"\"\"\")\" #G\"#c\"#qF&F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"\"\"*\"#O \"#%)\"$E\"F'F&F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"\"\"#5\"# X\"$?\"\"$5#\"$_#F'F&F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.\"\"\" \"#6\"#b\"$l\"\"$I$\"$i%F(F'F&F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6/\"\"\"\"#7\"#m\"$?#\"$&\\\"$#z\"$C*F(F'F&F%F$F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "60\"\"\"\"#8\"#y\"$'G\"$:(\"%(G\"\"%; " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Pascal's triangle .. 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We could construct our own binomial function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "A simpl e way to do this is to use the arrow notation " }{TEXT 0 2 "->" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "C2 := (n,r) -> n!/(r!*(n-r)!);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2Gf*6$%\"nG%\"rG6\"6$%)operatorG%&arrowGF)*&-%*f actorialG6#9$\"\"\"*&-F/6#9%F2-F/6#,&F1F2F6!\"\"F2F:F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The new binomia l function " }{TEXT 0 2 "C2" }{TEXT -1 60 " works fine for calculating a few rows of Pascal's triangle." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "n := 18:\nfor i from 1 to n \+ do\n print(seq(C2(i,r),r=0..i));\nend do;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 20 "The calculation of " }{XPPEDIT 18 0 "C(n,r)" "6#-%\"C G6$%\"nG%\"rG" }{TEXT -1 18 " from the formula " }{XPPEDIT 18 0 "n!/(r !*(n-r)!);" "6#*&-%*factorialG6#%\"nG\"\"\"*&-F%6#%\"rGF(-F%6#,&F'F(F, !\"\"F(F0" }{TEXT -1 44 " involves more arithmetic than is necessary. " }}{PARA 0 "" 0 "" {TEXT -1 43 "Let's construct a more efficient proc edure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Note the general scheme for constructing a procedure:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 27 "name of procedure := \+ proc( " }{TEXT -1 27 ". . arguments or input . . " }{TEXT 0 1 ")" }} {PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 0 5 "local" }{TEXT -1 63 " . . list of variables.. used internally in the procedure . . " }{TEXT 0 1 ";" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 " \+ . . . BODY OF PROCEDURE . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 260 73 "last statement givin g the value or Maple object returned by the procedure" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 0 9 "end proc;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 2 "C3" } {TEXT -1 42 " below evaluates the binomial coefficient:" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "C(n,r) =n!/(r!*(n-r)!) " "6#/ -%\"CG6$%\"nG%\"rG*&-%*factorialG6#F'\"\"\"*&-F+6#F(F--F+6#,&F'F-F(!\" \"F-F4" }{XPPEDIT 18 0 "`` = n*(n-1)*` . . . `*(n-r+2)*(n-r+1)/r!;" "6 #/%!G*.%\"nG\"\"\",&F&F'F'!\"\"F'%(~.~.~.~GF',(F&F'%\"rGF)\"\"#F'F',(F &F'F,F)F'F'F'-%*factorialG6#F,F)" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 62 "where n is a positive integer and r is a non-negative int eger." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " The numerator " }{XPPEDIT 18 0 "n*(n-1)*` . . . `*(n-r+2)*(n-r+1)" "6# *,%\"nG\"\"\",&F$F%F%!\"\"F%%(~.~.~.~GF%,(F$F%%\"rGF'\"\"#F%F%,(F$F%F* F'F%F%F%" }{TEXT -1 17 " and denominator " }{XPPEDIT 18 0 "r!;" "6#-%* factorialG6#%\"rG" }{TEXT -1 95 " are constucted simultaneously in the same loop, and the quotient is returned by the procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "It also makes us e of the fact that C(n,r) = C(n,n-r) to cut down on the number of arit hmetic operations required when r > " }{XPPEDIT 18 0 "n/2;" "6#*&%\" nG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "C3 := proc(n::posint,r::int eger)\n local num,denom,j;\n if r > n or r < 0 then return 0 end i f;\n if r > n/2 then return C3(n,n-r) end if;\n num := 1;\n deno m := 1;\n for j from 1 to r do\n num := num*(n-j +1);\n de nom := denom*j;\n end do;\n num/denom;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "C3(21,1 0);\nbinomial(21,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"';FN" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"';FN" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can construct Pascal's triangle w ith the new procedure " }{TEXT 0 2 "C3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "n := 18 :\nfor i from 1 to n do\n print(seq(C3(i,r),r=0..i));\nend do;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "A relation between bino mial coefficients .. 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The addition scheme which generates Pascal's triang le corresponds to a relationship between binomial coefficients, namely :" }}{PARA 256 "" 0 "" {TEXT -1 33 " C(n,r) + C(n,r+1) = C(n+1,r+1). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Here \+ are some numerical examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "alias(C=binomial);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"CG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C(20,7)+C(20,8);\nC(21,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"'!\\.#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"'!\\.#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "C(200,82)+C(200,83);\nC(201,83);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"en+?\\bg:u:hSXBnhG&[&*f?)\\jgJ9Xew& )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"en+?\\bg:u:hSXBnhG&[&*f?)\\jgJ9 Xew&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "C(2325,782)+C(2325,783);\nC(2326,783);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can make use th e formula C(n,r) + C(n,r+1) = C(n+1,r+1) to give a " }{TEXT 259 20 "re cursive definition" }{TEXT -1 162 " of binomial coefficients. The func tion is defined in terms of itself, but in a way that builds up rememb ered values in the form of the rows of Pascal's triangle." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "C4 : = proc(n::posint,r::integer)\n option remember;\n if r > n or r < \+ 0 then return 0 end if;\n if n < 2 then n else C4(n-1,r-1) + C4(n-1, r) fi;\nend proc:\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "C4(5,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "n := 18:\nfor i from 1 to n do\n print(seq(C4(i,r), r=0..i));\nend do;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 156 "When n is large the computation using the recursive de finition can be rather slow, because it has to build up many lower val ues to get at the desired value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "C4(500,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"ao!piL\"zKHB_#zi!>ylXD\")[5mrov " 0 "" {MPLTEXT 1 0 11 "C4(500,49);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"`o+&fj!GjJAh]ob-)z>*eZEbcGdEo^7 " 0 "" {MPLTEXT 1 0 194 "C5 := proc(n::posint,r::int eger)\n option remember;\n print(`C5(`||n||`,`||r||`)`);\n if r \+ > n or r < 0 then return 0 end if;\n if n < 2 then n else C5(n-1,r-1 ) + C5(n-1,r) end if;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "The function calls itself numerous times. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "C5(7,6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(7,6)G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(6,5)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(5,4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(4,3) G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(3,2)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(2,1)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(1,0) G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(1,1)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(2,2)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(1,2) G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(3,3)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(2,3)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(4,4) G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(3,4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(5,5)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(4,5) G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(6,6)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(C5(5,6)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Tasks 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the binomial expansion of each of the following. " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "(x-2)^3" "6#*$,&%\"xG\"\"\" \"\"#!\"\"\"\"$" }{TEXT -1 8 " (b) " }{XPPEDIT 18 0 "(2*u-3*v)^4" " 6#*$,&*&\"\"#\"\"\"%\"uGF'F'*&\"\"$F'%\"vGF'!\"\"\"\"%" }{TEXT -1 8 " \+ (c) " }{XPPEDIT 18 0 "(y-2)^5" "6#*$,&%\"yG\"\"\"\"\"#!\"\"\"\"&" } {TEXT -1 8 " (d) " }{XPPEDIT 18 0 "(2*s-t)^6" "6#*$,&*&\"\"#\"\"\"% \"sGF'F'%\"tG!\"\"\"\"'" }{TEXT -1 8 " (e) " }{XPPEDIT 18 0 "(x^2+1 )^8" "6#*$,&*$%\"xG\"\"#\"\"\"F(F(\"\")" }{TEXT -1 8 " (f) " } {XPPEDIT 18 0 "(c+d/2)^9" "6#*$,&%\"cG\"\"\"*&%\"dGF&\"\"#!\"\"F&\"\"* " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "(x-2)^3 = x^3-6*x^2+12*x-8" "6# /*$,&%\"xG\"\"\"\"\"#!\"\"\"\"$,**$F&F*F'*&\"\"'F'*$F&F(F'F)*&\"#7F'F& F'F'\"\")F)" }{TEXT -1 7 " (b) " }{XPPEDIT 18 0 "(2*u-3*v)^4 = 16*u^ 4-96*u^3*v+216*u^2*v^2-216*u*v^3+81*v^4" "6#/*$,&*&\"\"#\"\"\"%\"uGF(F (*&\"\"$F(%\"vGF(!\"\"\"\"%,,*&\"#;F(*$F)F.F(F(*(\"#'*F(*$F)F+F(F,F(F- *(\"$;#F(*$F)F'F(F,F'F(*(F7F(F)F(F,F+F-*&\"#\")F(*$F,F.F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "(y-2)^5 = y^5-10*y^4+40*y^3-80*y^2+80*y-32" "6#/*$,&%\"yG\"\"\"\"\"#!\"\"\"\"&, .*$F&F*F'*&\"#5F'*$F&\"\"%F'F)*&\"#SF'*$F&\"\"$F'F'*&\"#!)F'*$F&F(F'F) *&F6F'F&F'F'\"#KF)" }{TEXT -1 6 " (d) " }{XPPEDIT 18 0 "(2*s-t)^6 = 6 4*s^6-192*s^5*t+240*s^4*t^2-160*s^3*t^3+60*s^2*t^4-12*s*t^5+t^6" "6#/* $,&*&\"\"#\"\"\"%\"sGF(F(%\"tG!\"\"\"\"',0*&\"#kF(*$F)F,F(F(*(\"$#>F(* $F)\"\"&F(F*F(F+*(\"$S#F(*$F)\"\"%F(F*F'F(*(\"$g\"F(*$F)\"\"$F(F*F \+ " 0 "" {MPLTEXT 1 0 169 "(x-2)^3=expand((x-2)^3);\n(2*u-3*v)^4=expand( (2*u-3*v)^4);\n(y-2)^5=expand((y-2)^5);\n(2*s-t)^6=expand((2*s-t)^6); \n(x^2+1)^8=expand((x^2+1)^8);\n(c+d/2)^9=expand((c+d/2)^9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%\"xG\"\"\"\"\"#!\"\"\"\"$F(,**$)F'F+F (F(*&\"\"'F()F'F)F(F**&\"#7F(F'F(F(\"\")F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&*&\"\"#\"\"\"%\"uGF)F)*&\"\"$F)%\"vGF)!\"\"\"\"%F ),,*&\"#;F))F*F/F)F)*(\"#'*F))F*F,F)F-F)F.*(\"$;#F))F*F(F))F-F(F)F)*(F 8F)F*F))F-F,F)F.*&\"#\")F))F-F/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*$),&%\"yG\"\"\"\"\"#!\"\"\"\"&F(,.*$)F'F+F(F(*&\"#5F()F'\"\"%F(F**& \"#SF()F'\"\"$F(F(*&\"#!)F()F'F)F(F**&F8F(F'F(F(\"#KF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&*&\"\"#\"\"\"%\"sGF)F)%\"tG!\"\"\"\"'F),0*& \"#kF))F*F-F)F)*(\"$#>F))F*\"\"&F)F+F)F,*(\"$S#F))F*\"\"%F))F+F(F)F)*( \"$g\"F))F*\"\"$F))F+F>F)F,*(\"#gF))F*F(F))F+F9F)F)*(\"#7F)F*F))F+F5F) F,*$)F+F-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&*$)%\"xG\"\"#\" \"\"F+F+F+\"\")F+,4*$)F)\"#;F+F+*&F,F+)F)\"#9F+F+*&\"#GF+)F)\"#7F+F+*& \"#cF+)F)\"#5F+F+*&\"#qF+)F)F,F+F+*&F9F+)F)\"\"'F+F+*&F5F+)F)\"\"%F+F+ *&F,F+F(F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%\"cG\"\"\"* &\"\"#!\"\"%\"dGF(F(\"\"*F(,6*$)F'F-F(F(*&#F-F*F(*&)F'\"\")F(F,F(F(F(* (F-F()F'\"\"(F()F,F*F(F(*&#\"#@F*F(*&)F'\"\"'F()F,\"\"$F(F(F(*&#\"#jF5 F(*&)F'\"\"&F()F,\"\"%F(F(F(*&#FD\"#;F(*&)F'FIF()F,FGF(F(F(*&#F " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 \+ " }}{PARA 0 "" 0 "" {TEXT -1 65 "Find the first three terms in the fol lowing binomial expansions. " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " } {XPPEDIT 18 0 "(2*x+y)^12" "6#*$,&*&\"\"#\"\"\"%\"xGF'F'%\"yGF'\"#7" } {TEXT -1 8 " (b) " }{XPPEDIT 18 0 "(x-y)^15" "6#*$,&%\"xG\"\"\"%\"y G!\"\"\"#:" }{TEXT -1 8 " (c) " }{XPPEDIT 18 0 "(m^2-2*w^3)^11" "6# *$,&*$%\"mG\"\"#\"\"\"*&F'F(*$%\"wG\"\"$F(!\"\"\"#6" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 " Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a ) " }{XPPEDIT 18 0 "(2*x+y)^12 = 4096*x^12+24576*x^11*y+67584*x^10*y^2 +` . . . `;" "6#/*$,&*&\"\"#\"\"\"%\"xGF(F(%\"yGF(\"#7,**&\"%'4%F(*$F) F+F(F(*(\"&wX#F(*$F)\"#6F(F*F(F(*(\"&%enF(*$F)\"#5F(F*F'F(%(~.~.~.~GF( " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "(x-y)^15 = x^15-15*x^14*y+105*x^13*y ^2-` . . . `;" "6#/*$,&%\"xG\"\"\"%\"yG!\"\"\"#:,**$F&F*F'*(F*F'*$F&\" #9F'F(F'F)*(\"$0\"F'*$F&\"#8F'F(\"\"#F'%(~.~.~.~GF)" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " } {XPPEDIT 18 0 "(m^2-2*w^3)^11 = m^22-22*m^20*w^3+220*m^18*w^6-` . . . \+ `;" "6#/*$,&*$%\"mG\"\"#\"\"\"*&F(F)*$%\"wG\"\"$F)!\"\"\"#6,**$F'\"#AF )*(F2F)*$F'\"#?F)F,F-F.*(\"$?#F)*$F'\"#=F)F,\"\"'F)%(~.~.~.~GF." } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "(2*x+y)^12=e xpand((2*x+y)^12);\n(x-y)^15=expand((x-y)^15);\n(m^2-2*w^3)^11=expand( (m^2-2*w^3)^11);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*$),&*&\"\"#\" \"\"%\"xGF)F)%\"yGF)\"#7F),<*&\"%'4%F))F*F,F)F)*(\"&wX#F)F+F))F*\"#6F) F)*(\"&%enF))F+F(F))F*\"#5F)F)*(\"'SE6F))F+\"\"$F))F*\"\"*F)F)*(\"'?n7 F))F+\"\"%F))F*\"\")F)F)*(\"'w85F))F+\"\"&F))F*\"\"(F)F)*(\"&O\"fF))F+ \"\"'F))F*FOF)F)*(\"&W`#F))F+FKF))F*FIF)F)*(\"%?zF))F+FEF))F*FCF)F)*( \"%gF+)F)\"#5F+F+*(\"&SA%F+)F.\"#@F+)F)\"\")F+F0*(FV F+)F.\"#CF+)F)F " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(2*x+y)^12;\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),& *&\"\"#\"\"\"%\"xGF(F(%\"yGF(\"#7F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #,<*&\"%'4%\"\"\")%\"xG\"#7F&F&*(\"&wX#F&%\"yGF&)F(\"#6F&F&*(\"&%enF&) F,\"\"#F&)F(\"#5F&F&*(\"'SE6F&)F,\"\"$F&)F(\"\"*F&F&*(\"'?n7F&)F,\"\"% F&)F(\"\")F&F&*(\"'w85F&)F,\"\"&F&)F(\"\"(F&F&*(\"&O\"fF&)F,\"\"'F&)F( FJF&F&*(\"&W`#F&)F,FFF&)F(FDF&F&*(\"%?zF&)F,F@F&)F(F>F&F&*(\"%g " 0 "" {MPLTEXT 1 0 8 "2048*12;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&wX#" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________ ______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 28 "(a) Find the coefficient of " }{XPPEDIT 18 0 "w^3*y^5" "6#*&%\" wG\"\"$%\"yG\"\"&" }{TEXT -1 21 " in the expansion of " }{XPPEDIT 18 0 "(w+y)^8" "6#*$,&%\"wG\"\"\"%\"yGF&\"\")" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) Find the coefficient of " }{XPPEDIT 18 0 "a ^6*z^3" "6#*&%\"aG\"\"'%\"zG\"\"$" }{TEXT -1 21 " in the expansion of \+ " }{XPPEDIT 18 0 "(a+z)^9" "6#*$,&%\"aG\"\"\"%\"zGF&\"\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "(c) Find the coefficient of " } {XPPEDIT 18 0 "b^8*a^5;" "6#*&%\"bG\"\")%\"aG\"\"&" }{TEXT -1 21 " in \+ the expansion of " }{XPPEDIT 18 0 "(b-2*a)^13" "6#*$,&%\"bG\"\"\"*&\" \"#F&%\"aGF&!\"\"\"#8" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "(d) Find the coefficient of the middle term in " }{XPPEDIT 18 0 "(a+b )^16;" "6#*$,&%\"aG\"\"\"%\"bGF&\"#;" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "C[8,5] = matrix([[8], [5]]);" "6#/&%\"CG6$\"\")\"\"&-%'matrixG6# 7$7#F'7#F(" }{XPPEDIT 18 0 "``=56" "6#/%!G\"#c" }{TEXT -1 6 " (b) " } {XPPEDIT 18 0 "C[9,3] = matrix([[9], [3]])" "6#/&%\"CG6$\"\"*\"\"$-%'m atrixG6#7$7#F'7#F(" }{XPPEDIT 18 0 "``=84" "6#/%!G\"#%)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) " }{XPPEDIT 18 0 "matrix([[13], [5]])*(-2)^5=-41184" "6#/*&-%'matrixG6#7$7#\"#87#\"\"&\"\"\"*$,$\"\"# !\"\"F,F-,$\"&%=TF1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "C[16,8] = matrix([[16], [8]]);" "6#/&%\"CG6$\"#;\" \")-%'matrixG6#7$7#F'7#F(" }{XPPEDIT 18 0 "``=16!/(8!*`.`*8!)" "6#/%!G *&-%*factorialG6#\"#;\"\"\"*(-F'6#\"\")F*%\".GF*-F'6#F.F*!\"\"" } {XPPEDIT 18 0 "``=16*`.`*15*`.`*14*`.`*13*`.`*12*`.`*11*`.`*10*`.`*9/( 8*`.`*7*`.`*6*`.`*5*`.`*4*`.`*3*`.`*2*`.`*1)" "6#/%!G*B\"#;\"\"\"%\".G F'\"#:F'F(F'\"#9F'F(F'\"#8F'F(F'\"#7F'F(F'\"#6F'F(F'\"#5F'F(F'\"\"*F'* @\"\")F'F(F'\"\"(F'F(F'\"\"'F'F(F'\"\"&F'F(F'\"\"%F'F(F'\"\"$F'F(F'\" \"#F'F(F'F'F'!\"\"" }{XPPEDIT 18 0 "``= 12870" "6#/%!G\"&qG\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________ ______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 18 "Find the value of " }{XPPEDIT 18 0 "Sum(C[12,j],j=0..12)" "6#-%$Su mG6$&%\"CG6$\"#7%\"jG/F*;\"\"!F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(C[12,j],j = 0 .. 12)=(1+1)^12" "6#/-%$SumG6$&%\"CG6$\"#7%\"jG/F+; \"\"!F**$,&\"\"\"F1F1F1F*" }{XPPEDIT 18 0 "``=2^12" "6#/%!G*$\"\"#\"#7 " }{XPPEDIT 18 0 "``=4096" "6#/%!G\"%'4%" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Sum(b inomial(12,j),j = 0 .. 12);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$-%)binomialG6$\"#7%\"jG/F*;\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%'4%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 " __________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "T asks 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 27 " (a) F ind the expansion of " }{XPPEDIT 18 0 "(x+y)^10;" "6#*$,&%\"xG\"\"\"% \"yGF&\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 " (b) Use th e Maple commands " }{TEXT 0 3 "seq" }{TEXT -1 5 " and " }{TEXT 0 8 "bi nomial" }{TEXT -1 60 " to obtain the sequence of coefficients in the e xpansion of " }{XPPEDIT 18 0 "(x+y)^10;" "6#*$,&%\"xG\"\"\"%\"yGF&\"#5 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "_____________________ _____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 126 "(a) Set up and execut e the following Maple commands which add up the binomial coefficients \+ in the10th row of Pascals triangle." }}{PARA 0 "" 0 "" {TEXT 261 48 "i := 'i':\nSum(binomial(10,i),i=0..10);\nvalue(%);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "(b) Set up and execute th e following Maple commands. What do these commands do?" }}{PARA 0 "" 0 "" {TEXT 261 47 "n := 'n': \nSum(binomial(n,i),i=0..n);\nvalue(%);" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "(c) Ex plain the results in (a) and (b). " }{TEXT 259 4 "Hint" }{TEXT -1 28 ": Consider the expansion of " }{XPPEDIT 18 0 "(a+b)^n;" "6#),&%\"aG\" \"\"%\"bGF&%\"nG" }{TEXT -1 31 " with a and b both equal to 1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "_________ _________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 42 "(a) Make a list of 1 1 points of the form: " }{XPPEDIT 18 0 "[i, binomial(10,i)/(2^10)];" " 6#7$%\"iG*&-%)binomialG6$\"#5F$\"\"\"*$\"\"#F)!\"\"" }{TEXT -1 75 ", f or i from 0 to 10, and assign this list of points to the variable \"pt s\"." }}{PARA 0 "" 0 "" {TEXT -1 53 "(b) Plot the points obtained in ( a). Use the option " }{TEXT 0 11 "style=point" }{TEXT -1 15 " for the Maple " }{TEXT 0 4 "plot" }{TEXT -1 9 " command." }}{PARA 0 "" 0 "" {TEXT -1 61 "(c) Repeat (a) and (b) with a list of 61 points of the fo rm: " }{XPPEDIT 18 0 "[i, binomial(60,i)/(2^60)];" "6#7$%\"iG*&-%)bino mialG6$\"#gF$\"\"\"*$\"\"#F)!\"\"" }{TEXT -1 22 ", for i from 0 to 60. " }}{PARA 0 "" 0 "" {TEXT -1 36 "(d) Plot the graph of the function \+ " }{XPPEDIT 18 0 "f(x) = exp(-(x-30)^2/30)/sqrt(30*Pi);" "6#/-%\"fG6#% \"xG*&-%$expG6#,$*&,&F'\"\"\"\"#I!\"\"\"\"#F0F1F1F/-%%sqrtG6#*&F0F/%#P iGF/F1" }{TEXT -1 26 ", for x between 0 and 60. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "(e) How are the points pl otted in (c) related to the graph of the function in (d)? " }}{PARA 0 "" 0 "" {TEXT -1 85 " You can plot the points and curve on the sam e graph by using the command . . . ." }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{TEXT 261 76 "plot([pts,f(x)],x=0..60,style=[point,line], symbo l=CIRCLE,color=[blue,red]);" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "(f) Describe the main features of the graph obt ained in (d)." }}{PARA 0 "" 0 "" {TEXT -1 34 "________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q4 \+ " }}{PARA 0 "" 0 "" {TEXT -1 51 "(a) Setup and execute the following \+ Maple commands:" }}{PARA 0 "" 0 "" {TEXT 261 68 "a := 'a': b := 'b':\n Sum(binomial(8,i)*a^(8-i)*b^i,i=0..8);\nvalue(%);" }}{PARA 0 "" 0 "" {TEXT -1 55 "(b) Identify the result in (a) as a binomial expansion." }}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictu res" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Binomial product picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 683 "ch := [`(`, `a`,`+`,`b`,`)`,`(`,`c`,`+`,`d`,`)`,` `,`=`,` `,\n `a c`,` `,`+`,` \+ `,`a d`,` `,`+`,` `,`b c`,` `,`+`,` `,`b d`]:\nt1 := plots[textplot]([ seq([i,0,ch[i]],i=1..nops(ch))]):\np1 := plot([[2.07,1],[2.07,2],[7,2] ,[7,1],[7,2],[14,2],[14,1]],thickness=2,color=red):\np2 := plot([[2,1] ,[2,3],[7,3],[9,3],[9,1],[9,3],[18,3],[18,1]],\n thickness= 2,color=COLOR(RGB,.8,0,.6)):\np3 := plot([[4.07,-1],[4.07,-2],[7,-2],[ 7,-1],[7,-2],[22,-2],[22,-1]],\n thickness=2,color=COLOR(RG B,0,.6,0)):\np4 := plot([[4,-1],[4,-3],[9,-3],[9,-1],[9,-3],[26,-3],[2 6,-1]],\n thickness=2,color=COLOR(RGB,0,.2,1)):\nplots[disp lay]([t1,p1,p2,p3,p4],view=[0..26,-3..3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 369 "p1 := pl ot([[[0,0],[6,0]],[[0,0],[0,4]],\n[[6,0],[6,4]],[[0,4],[6,4]],[[0,1],[ 6,1]],\n[[4,0],[4,4]]],color=COLOR(RGB,0.1,0.6,0.1)):\nt1 := plots[tex tplot]([[2,2.5,`area = ac`],[2,.5,`area = bc`],\n [5,2.5,`area = ad`] ,[5,.5,`area = bd`]],color=blue):\nt2 := plots[textplot]([[-0.15,0.5,` b`],[-0.15,2.5,`a`],[2,4.18,`c`],[5,4.18,`d`]]):\nplots[display]([p1,t 1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "A ddition scheme for coefficients " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "t1 := plots[textplot]([[0,5 ,`p`],[2,5,`q`],[2,0,`p + q`]],font=[HELVETICA,10]):\np1 := plottools[ arrow]([.5,5],[1.5,5],0,.6,.23,arrow,thickness=1,color=black):\np2 := \+ plottools[arrow]([2,3.6],[2,1],0,.2,.23,arrow,thickness=1,color=black) :\nplots[display]([p1,p2,t1],view=[0..2.5,-1..5.5],axes=none);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 595 "ch1 := [`0`,` `,`1`,` `,`3`,` `,`3`,` `,`1`,` `,`0`]:\nch2 := [`1 `,` `,`4`,` `,`6`,` `,`4`,` `,1]:\nt1 := plots[textplot]([seq([i,4,ch1 [i]],i=1..nops(ch1))]):\nt2 := plots[textplot]([seq([i+2,0,ch2[i]],i=1 ..nops(ch2))]):\nclrs := [COLOR(RGB,0,.8,0),COLOR(RGB,.4,0,.8),COLOR(R GB,1,.1,0),\n COLOR(RGB,.7,.7,0),COLOR(RGB,.65,.2,.2)]:\np1 := seq (plottools[arrow]([2*i-.5,4],[2*i+.5,4],0,.5,.2,arrow,\n thickness=2 ,color=clrs[i]),i=1..5):\np2 := seq(plottools[arrow]([1+2*i,3],[1+2*i, 1],0,.25,.2,arrow,\n thickness=2,color=clrs[i]),i=1..5):\nplots[disp lay]([p1,p2,t1,t2],view=[0..12,-1..5],axes=none);\n" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }