Find the coefficients of `x^(50)` in the expression `(1+x)^(1000)+2x(1+x)^(999)+3x^2(1+x)^(998)++1001 x^(1000)` .

Options

A. `""^(1000)C_(50)`
B. `""^(1001)C_(50)`
C. `""^(1002)C_(50)`
D. `2^(1001)`

Answer

C

Explanation

Let, `S=(1+x)^(1000)+2x(1+x)^(999)+3x^(2)(1+x)^(998)+.....+1001x^(1000)`
`(x)/(1+x)S=x(1+x)^(999)+2x^(2)(1+x)^(998)+......+ 1000x^(1000)+(1001x^(1001))/(1+x)`
Subtract above equations,
`(1+(x)/(1+x))S=(1+x)^(1000)+(1+x)^(999)+x^(2)(1+x)^(998)+.......+ x^(1000)- (1001x^(1001))/(1+x)`
`rArr S=(1+x)^(1001)+x(1+x)^(1000)+x^(2)(1+x)^(9999)+......+ x^(1000)(1+x)-1001x^(1001)`
`=((1+x)^(1001)[((x)/(1+x))^(1001)-1])/((x)/(1+x)-1)-1001x^(1001)`
[sum of G.P.] `=(1+x)^(1002)-x^(1002)-1002x^(1001) therefore "coefficient of " x^(50)" in " S ="coefficient of " x^(50)` in
`[(1+x)^(1002)-x^(1002)-1002x^(1001)]=""^(1002)C_(50)`