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 $\left(1 - 2x + 3x^{2}\right) \left(1 + x\right)^{11 }$
$= \left(1 - 2x + 3x^{2}\right)$
$\times \left(1+\,{}^{11}C_{1}x +\,{}^{11}C_{2}x^{2}+\ldots\,{}^{11}C_{9}x^{9}+\,{}^{11}C_{10}x^{10}+x^{11}\right)$
$\therefore $ Coefficient of $x^{11} = 1 \times 1 - 2 \times \,{}^{11}C_{10 }+ 3\times \,{}^{11}C_{9}$
$= 1-2 \times 11 + 3 \times\frac{11\times 10}{2!} = 144$