Question

Coefficient of $x^{11}$in the expansion of $(1+x^2{)}^4(1+x^3{)}^7(1+x^4{)}^{12}$ is

• A. 1051
• B. 1106
• C. 1113
• D. 1120

Answer 1

The correct option is C 1113

Coeff. of $x^{11}$ in exp. of $(1+x^2{)}^4(1+x^3{)}^7(1+x^4{)}^{12}(\text{Coeff.of}{x}^a)\times (\text{Coeff.of}{x}^b)\times (\text{Coeff.of}{x}^c)\text{Such that}a+b+c=11\text{Here}a=2m,b=3n,c=4p\therefore 2m+3n+4p=11\text{Case I}:m=0,n=1,p=2\text{Case II}:m=1,n=3,p=0\text{Case III}:m=2,n=1,p=1\text{Case IV}:m=4,n=1,P=0\therefore \text{Required coeff.}={}^4{C}_0\times {{}^{7}C}_1\times {}^{12}{C}_2+{}^4{C}_1\times {}^7{C}_3\times {}^{12}{C}_0+{}^4{C}_2\times {}^7{C}_1\times {}^{12}{C}_1+{}^4{C}_4\times {}^7{C}_1\times {}^{12}{C}_0=462+140+504+7=1113$