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 `(n(n+1)(n+2)(n+r))/(r !)` `((n+1)(n+2)(n+r))/(r !)` `(n(n+1)(n+2)(n+r-1))/(r !)` none of these

Options

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

Answer

b

Explanation

We have,
`(1 -x)^(-n) = a_(0) + a_(1) x + a_(2) x^(2) +…+ a_(r) x^® +…`
and, `(1 - x )^(-1) = 1 + x + x^(2) + x^(3) + …+ x^(r) + …`
`therefore 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))`
=` ((n+ 1) (n+2) ...(n+r))/(r!)` .