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)^(1000) + x(1+x)^(1000) +x(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) :. "coefficient of " x^50" in "S = "coefficient of "x^50 " in "`
`[(1+x)^(1002)-x^1002- 1002x^1001]=""^1002C_50`