The coefficient of `1//x` in the expansion of `(1+x)^n(1+1//x)^n` is `(n !)/((n-1)!(n+1)!)` b. `((2n)!)/((n-1)!(n+1)!)` c. `((2n)!)/((2n-1)!(2n+1)!)` d. none of these

Explanation

Given
`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 ` (1+frac{1}{x})^{n}+` Coefficient of `x` in `(1+x)^{n} times` Coefficient of
`frac{1}{x^{2}}` in `(1+frac{1}{x})^{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} `

`=frac{(2 n) !}{[(n-1) !(n+1) !]}`