The coefficient of `x^(n)` in the expansion of `(1+x)(1-x)^(n)` is

Options

A. `(n(n^(2)+2)(3n+1))/(24)`
B. `(n(n^(2)-1)(3n+2))/(24)`
C. `(n(n^(2)+1)(3n+4))/(24)`
D. None of these

Answer

B

Explanation

Let `f(x)=(x-1)(x-2)(x-3)"…."(x-n)`
`=x^(n)-S_(1)x^(n-1)+S_(2)x^(n-2)-"..."+(-1)^(n)(1*2*3*"..."n)`
So, coefficient of `x^(n-2)` in `f(x)=S_(2)=(1*2*3*"..."n)`
=Sum of product of first n natural number taken 2 at time
`=(1)/(2)[(1+2+"......"+n)^(2)-(1^(2)+2^(2)+"....."+n^(2))]`
`=(1)/(2)[{(n(n+1))/(2)}^(2)-(n(n+1)(2n+1))/(6)]`
`=(1)/(2)*(n(n+1))/(2)[(n(n+1))/(2)-(2n+1)/(3)]`
`=(1)/(2)*(n(n+1))/(2)[(3n^(2)+3n-4n-2)/(6)]`
`=(n(n+1)(3n^(2)+3n-4n-2))/(24)=(n(n+1)(3n+2)(n-1))/(24)`
`=(n(n^(2)-1)(3n+2))/(24)`.