The coefficient of

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 $\left(1+\frac{1}{x}\right)^{n}$

$={ }^{n} C_{0} \times{ }^{n} C_{1}+{ }^{n} C_{1} \times{ }^{n} C_{2}$

$=n+n \times \frac{n !}{2(n-2) !}$

$=n+n \frac{n(n-1)}{2}$

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