Question

The coefficient of $\frac{1}{x}$ in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$ is

• A. $\frac{n!}{(n-1)!(n+1)!}$
• B. $\frac{\left(2n\right)!}{(n-1)!(n+1)!}$
• C. $\frac{\left(2n\right)!}{(2n-1)!(2n+1)!}$
• D. $\frac{2n!}{(n-2)!(n+2)!}$

Answer 1

The correct option is B $\frac{\left(2n\right)!}{(n-1)!(n+1)!}$

Coefficient of $x^{-1}$ in $(1+x{)}^n{(1+\frac{1}{x})}^n$
$=$ Coefficient of $x^{-1}$ in $x^{-n}(1+x{)}^{2n}$
$=$ Coefficient of $x^{n-1}$ in $(1+x{)}^{2n}$
Now as we know that coefficient of $x^r$ in expansion of $(1+x{)}^n$ is ${}^n{C}_r$
Hence required coefficient is
$={}^{2n}{C}_{n-1}$
$=\frac{\left(2n\right)!}{(n-1)!(n+1)!}$