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

Options

A. `(n!)/((n-1)!(n+1)!)`
B. `(2n!)/((n-1)!(n+1)!)`
C. `(n!)/((2n-1)!(1n+1)!)`
D. `(2n!)/((2n-1)!(1n+1)!)`

Answer

b

Explanation

Now , ` (1 + x)^(n) (1+ (1)/(x))^(n) = ((1 + x)^(2n))/(x^(n))`
` therefore ` Coefficient of ` x^(-1) " in" (1 + x)^(n) (1+ (1)/(x))^(n)`
= Coefficient of ` x^(n-1)" in " (1 + x)^(2n) = ""^(2n)C_(n-1) = ((2n)!)/((n-1)!(n+1)!)` .