The coefficient of `x^n` in `(1+x)^(101)(1-x+x^2)^(100)` is nonzero, then `n` cannot be of the form `3r+1` b. `3r` c. `3r+2` d. none of these

Options

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

Answer

C

Explanation

We havec
`(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){C_(0) + C_(1)x^(3)+C_(2)x^(6)+"…."+C_(100)x^(300)}`, where `C_(r) = .^(100)C_(r)`
`= (1+x)underset(r=0)overset(n)sum.^(n)C_(r)x^(3r)=underset(r=0)overset(n)sum.^(n)C_(r)x^(3r)+underset(r=0)overset(n)sum.^(n)C_(r)x^(3r+1)`
Hence, there will be no term containing `3r+2`.