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Question Number 122250 by mr W last updated on 15/Nov/20

Answered by mr W last updated on 15/Nov/20

using generating function method  the number of ways is the coefficient  of term x^(10)  in the expansion  (C_1 ^(15) x+C_2 ^(15) x^2 +...+C_(15) ^(15) x^(15) )×  (C_1 ^(12) x+C_2 ^(12) x^2 +...+C_(12) ^(12) x^(12) )×  (C_1 ^(10) x+C_2 ^(10) x^2 +...+C_(10) ^(10) x^(10) )×  (C_1 ^5 x+C_2 ^5 x^2 +...+C_5 ^5 x^5 )  which is 1 032 675 125.    general form  GF=Π_(i=1) ^m (Σ_(k=1) ^n_i  C_k ^n_i  x^k )

$${using}\:{generating}\:{function}\:{method} \\ $$$${the}\:{number}\:{of}\:{ways}\:{is}\:{the}\:{coefficient} \\ $$$${of}\:{term}\:{x}^{\mathrm{10}} \:{in}\:{the}\:{expansion} \\ $$$$\left({C}_{\mathrm{1}} ^{\mathrm{15}} {x}+{C}_{\mathrm{2}} ^{\mathrm{15}} {x}^{\mathrm{2}} +...+{C}_{\mathrm{15}} ^{\mathrm{15}} {x}^{\mathrm{15}} \right)× \\ $$$$\left({C}_{\mathrm{1}} ^{\mathrm{12}} {x}+{C}_{\mathrm{2}} ^{\mathrm{12}} {x}^{\mathrm{2}} +...+{C}_{\mathrm{12}} ^{\mathrm{12}} {x}^{\mathrm{12}} \right)× \\ $$$$\left({C}_{\mathrm{1}} ^{\mathrm{10}} {x}+{C}_{\mathrm{2}} ^{\mathrm{10}} {x}^{\mathrm{2}} +...+{C}_{\mathrm{10}} ^{\mathrm{10}} {x}^{\mathrm{10}} \right)× \\ $$$$\left({C}_{\mathrm{1}} ^{\mathrm{5}} {x}+{C}_{\mathrm{2}} ^{\mathrm{5}} {x}^{\mathrm{2}} +...+{C}_{\mathrm{5}} ^{\mathrm{5}} {x}^{\mathrm{5}} \right) \\ $$$${which}\:{is}\:\mathrm{1}\:\mathrm{032}\:\mathrm{675}\:\mathrm{125}. \\ $$$$ \\ $$$${general}\:{form} \\ $$$${GF}=\underset{{i}=\mathrm{1}} {\overset{{m}} {\prod}}\left(\underset{{k}=\mathrm{1}} {\overset{{n}_{{i}} } {\sum}}{C}_{{k}} ^{{n}_{{i}} } {x}^{{k}} \right) \\ $$

Commented by mr W last updated on 15/Nov/20

if we choose totally 4 books, the  answer is 15×12×10×5=9 000.  (=coef. of term x^4 ) ✓    if we choose totally 5 books, the  answer is  C_2 ^(15) ×12×10×5+C_2 ^(12) ×15×10×5+  C_2 ^(10) ×15×12×5+C_2 ^5 ×15×12×10  =171 000.  (=coef. of term x^5 ) ✓

$${if}\:{we}\:{choose}\:{totally}\:\mathrm{4}\:{books},\:{the} \\ $$$${answer}\:{is}\:\mathrm{15}×\mathrm{12}×\mathrm{10}×\mathrm{5}=\mathrm{9}\:\mathrm{000}. \\ $$$$\left(={coef}.\:{of}\:{term}\:{x}^{\mathrm{4}} \right)\:\checkmark \\ $$$$ \\ $$$${if}\:{we}\:{choose}\:{totally}\:\mathrm{5}\:{books},\:{the} \\ $$$${answer}\:{is} \\ $$$${C}_{\mathrm{2}} ^{\mathrm{15}} ×\mathrm{12}×\mathrm{10}×\mathrm{5}+{C}_{\mathrm{2}} ^{\mathrm{12}} ×\mathrm{15}×\mathrm{10}×\mathrm{5}+ \\ $$$${C}_{\mathrm{2}} ^{\mathrm{10}} ×\mathrm{15}×\mathrm{12}×\mathrm{5}+{C}_{\mathrm{2}} ^{\mathrm{5}} ×\mathrm{15}×\mathrm{12}×\mathrm{10} \\ $$$$=\mathrm{171}\:\mathrm{000}. \\ $$$$\left(={coef}.\:{of}\:{term}\:{x}^{\mathrm{5}} \right)\:\checkmark \\ $$

Commented by mr W last updated on 15/Nov/20

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