Question

The coefficient of $x^{20}$ in the expansion of $(1+x^2{)}^{40}.{(x^2+2+\frac{1}{x^2})}^{-5}$ is

• A. ${}^{30}{C}_{10}$
• B. ${}^{30}{C}_{25}$
• C. $1$
• D. none of these

Answer 1

The correct option is B ${}^{30}{C}_{25}$

Given expansion is
$(1+x^2{)}^{40}.{(x^2+2+\frac{1}{x^2})}^{-5}$
$=(1+x^2{)}^{40}.{(x+\frac{1}{x})}^{-10}$
$=\frac{(1+x^2{)}^{40}(1+x^2{)}^{-10}}{x^{-10}}$
$=\frac{(1+x^2{)}^{30}}{x^{-10}}$
$=x^{10}(1+x^2{)}^{30}$
Now, coefficient of $x^{10}$ in the expansion of $(1+x^2{)}^{30}$ is
${}^{30}{C}_5{=}^{30}{C}_{25}$
So, coefficient of $x^{20}$ in the given expansion is ${}^{30}{C}_{25}$