The coefficient of `x^n` in `(1+x)^(101)(1-x+x^2)^(100)` is non zero, then `n` cannot be of the form

Options

A. `3lambda + 1`
B. `3lambda`
C. `3lambda+2`
D. none of these

Answer

C

Explanation

`because(1+x)^(101) (1-x+x^(2))^(100)=(1+x)((1+x)(1-x+x^(2)))^(100)`
`= (1+x)(1+x^(3))^(100)`
`=(1+x)(1+^(100)C_(1)x^(3)+^(100)C_(2)x^(6)+C_(3)x^(9)+...+...+^(100)C_(10)x^(100))`
Clearly, in this expressinon `x^(3)` will present if `n= 3 lambda ` or
`n=3lambda + 1.` So, n cannot be of the form `3lambda + 2.`