Question

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

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

Answer 1

Answer: (C)

$2x_1+3x_2+4x_3=11$
Possibilities are $(0,1,2);(1,3,0);(2,1,1);(4,1,0)$.
$\therefore$ Required coefficients
$=\left({}^4{C}_0\times {}^7{C}_1\times {}^{12}{C}_2\right)+\left({}^4{C}_1\times {}^7{C}_3\times {}^{12}{C}_0\right)+\left({}^4{C}_2\times {}^7{C}_1\times {}^{12}{C}_1\right)+\left({}^4{C}_4\times {}^7{C}_1\times 1\right)$
$=(1\times 7\times 66)+(4\times 35\times 1)+(6\times 7\times 12)+(1\times 7)$
$=462+140+504+7=1113$