Question

The coefficient of $x^2$ in the expansion of $\left(1-x+2x^2\right){\left(x+\frac{1}{x}\right)}^{10}$ is

• A. $210$
• B. $714$
• C. $504$
• D. $240$

Answer 1

Answer: (B)

The required coefficient is the coefficient of $x^{12}$ in $\left(1-x+2x^2\right){\left(x^2+1\right)}^{10}$
$=\left(1-x+2x^2\right)\left(1+{}^{10}{C}_1{x}^2+{}^{10}{C}_2{x}^4+\dots \dots +{}^{10}{C}_9{x}^{18}+x^{20}\right)$
Coefficient of $x^{12}$ is
$1\times {}^{10}{C}_6+2\times {}^{10}{C}_5=\frac{10\times 93\times 8\times 7}{4\times 3\times 2}+\frac{2\times 10\times \times 93\times 82\times 7\times 6}{5\times 4\times 3\times 2}$
$=210+504$
$=714$