Question:
The coefficient of $\frac{1}{x}$ in the expansion of $(1+x)^{n}\left(1+\frac{1}{x}\right)^{n}$ 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
Solution:
(b) $\frac{(2 n) !}{[(n-1) !(n+1) !]}$
Coefficient of $\frac{1}{x}$ in the given expansion $=$ Coefficient of 1 in $(1+x)^{n} \times$ Coefficient of $\frac{1}{x}$ in $\left(1+\frac{1}{x}\right)^{n}+$ Coefficient of $x$ in $(1+x)^{n} \times$ Coefficient of $\frac{1}{x^{2}}$ in $\left(1+\frac{1}{x}\right)^{n}$
$={ }^{n} C_{0} \times{ }^{n} C_{1}+{ }^{n} C_{1} \times{ }^{n} C_{2}$
$=n+n \times \frac{n !}{2(n-2) !}$
$=n+n \frac{n(n-1)}{2}$