If `(1-x)^(-n)=a_0+a_1x+a_2x^2++a_r x^r+ ,t h e na_0+a_1+a_2++a_r` is equal to (a)`(n(n+1)(n+2)(n+r))/(r !)` (b)`((n+1)(n+2)(n+r))/(r !)` (c)`(n(n+1)(n+2)(n+r-1))/(r !)` (d)none of these

Options

A. `(n(n+1)(n+2)"…."(n+r))/(r!)`
B. `((n+1)(n+2)"…."(n+r))/(r!)`
C. `(n(n+1)(n+2)"….."(n+r-1))/(r!)`
D. none of these

Answer

B

Explanation

We have,
`a_(0) + a_(1) + a_(2) +"……"+a_(r)`
= Coefficient of `x^(r)` in the product of the two series
= Coefficient of `x^(r)` in `(1-x)^(-n)(1-x)^(-1)`
= Coefficient of `x^(r)` in `(1-x)^(-(n+1))`
`= ((n+1)(n+2)"…."(n+r))/(r!) = .^(n+r)C_(n)`