Question

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

• A. ${}^{10}{C}_5$
• B. ${}^{10}{C}_6$
• C. ${}^{11}{C}_5$
• D. ${}^{17}{C}_7$

Answer 1

Answer: (C)

We have, $(1-x{)}^2{\left(x+\frac{1}{x}\right)}^{10}$ $\begin{array}{l}=\left(1-2x+x^2\right)\left(x^{10}+{}^{10}{C}_1{x}^9\cdot \frac{1}{x}+{}^{10}{C}_2{x}^8\times \frac{1}{x^2}+\dots +\frac{1}{x^{10}}\right)\\ =\left(1-2x+x^2\right)\left(x^{10}+{}^{10}{C}_1{x}^8+{}^{10}{C}_2{x}^6+\dots +{}^{10}{C}_9\times \frac{1}{x^8}+\frac{1}{x^{10}}\right)\end{array}$
So, the term independent of $x$ is
$\begin{array}{l}{}^{10}{C}_5+(-2)\times 0+{}^{10}{C}_6={}^{10}{C}_5+{}^{10}{C}_6\\ ={}^{11}{C}_5\end{array}$