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The coefficient of x^n in (1+x)^(101)(1-...

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

A

`3lambda + 1`

B

`3lambda`

C

`3lambda+2`

D

`4lambda+1`

Text Solution

Verified by Experts

The correct Answer is:
c
`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.`
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