Question

Constant term in the expansion of ${(x-\frac{1}{x})}^{10}$ is

• A. $152$
• B. $-152$
• C. $-252$
• D. $252$

Answer 1

Answer: (C)

$-252$

${(x-\frac{1}{x})}^{10}$
$T_{r+1}{=}^{10}{C}_r(x{)}^{10-r}{\left(\frac{-1}{x}\right)}^r$

${=}^{10}{C}_r{x}^{10-r}\frac{(-1{)}^r}{x^r}$

${=}^{10}{C}_r(-1{)}^r{x}^{10-2r}$
For constant term $10-2r=0$
$\Rightarrow r=5$
${}^{10}{C}_r(-1{)}^5={-}^{10}{C}_5=\frac{-(10!)}{5!5!}$
$=-252$