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

Answer: (B)

$^{30}C_{25}$

Answer 2

Expression = $= \left(1+x^{2}\right)^{40} .\left(x+\frac{1}{x}\right)^{-10}$
$=\left(1+x^{2}\right)^{30} .x^{10}$
The coefficient of $x^{20}$ in $x^{10}\left(1+x^{2}\right)^{30}$
$=$ the coefficient of $x^{10}$ in $\left(1+x^{2}\right)^{30}$
$= \,^{30} C_{5} =\,^{30}C_{30-5} = \,^{30}C_{25}$