Question

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

• A. $\frac{ n !}{( n -1) !( n +1) !}$
• B. $\frac{2 n !}{(n-1) !(n+1) !}$
• C. $\frac{(2 n) !}{(2 n-1) !(2 n+1) !}$
• D. none of these

Answer 1

Answer: (B)

Coefficient of $x ^{-1}$ in $(1+ x )^{ n }\left(1+\frac{1}{ x }\right)^{ n }$
$=$ Coefficient of $x ^{-1}$ in $\frac{(1+ x )^{2 n }}{ x ^{ n }}$
$=$ Coefficient of $x ^{ n -1}$ in $(1+ x )^{2 n }$
$={ }^{2 n } C _{ n -1}=\frac{(2 n ) !}{( n -1) !( n +1) !}$