Question

The coeffcient of $x^{30}$ in the expansion of $(1+x^2{)}^{60}(x^2+2+\frac{1}{x^2}{)}^{-10}$ is

• A. ${}^{40}{\text{C}}_{10}$
• B. $1$
• C. ${}^{40}{\text{C}}_{20}$
• D. ${}^{40}{\text{C}}_5$

Answer 1

Answer: (D)

${}^{40}{\text{C}}_5$

The above question can be written as
$(1+x^2{)}^{60}((x+\frac{1}{x}{)}^2{)}^{-10}$
$=(1+x^2{)}^{60}(x+\frac{1}{x}{)}^{-20}$
$=(1+x^2{)}^{60}(\frac{1+x^2}{x}{)}^{-20}$
$=x^{20}(1+x^2{)}^{40}$
Hence coefficient of $x^{30}=x^{20}.x^{10}$ will be
$=$coefficient of $x^{10}$ in $(1+x^2{)}^{40}$
$={}^{40}{C}_5$