Question

Find the coefficient of $x^{11}$ in the expansion of $(1-2x+3x^2)(1+x{)}^{11}$.

• A. $132$
• B. $169$
• C. $144$
• D. $184$

Answer 1

Answer: (C)

We have $\left(1-2x+3x^2\right){\left(1+x\right)}^{11}$
$=\left(1-2x+3x^2\right)$
$\times \left(1+{}^{11}{C}_{1}x+{}^{11}{C}_2{x}^2+\dots {}^{11}{C}_9{x}^9+{}^{11}{C}_{10}{x}^{10}+x^{11}\right)$
$\therefore$ Coefficient of $x^{11}=1\times 1-2\times {}^{11}{C}_{10}+3\times {}^{11}{C}_9$
$=1-2\times 11+3\times \frac{11\times 10}{2!}=144$