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The coefficient of x^3 in the expansion ...

The coefficient of `x^3` in the expansion of `(1-x+x^2)^5` is

A

`-83`

B

`0`

C

`.^(30)C_(10)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
D
`(x-2)^(5) (x+1)^(5)`
`= [.^(5)C_(0)x^(5) - .^(5)C_(1)x^(4)xx2+"….."][.^(5)C_(0) + .^(5)C_(1)x +"….."]`
`rArr`Coefficient of `x^(5)`
`=.^(5)C_(0).^(5)C_(5) - .^(5)C_(1) xx 2 xx .^(5)C_(4) + .^(5)C_(2) xx 2^(2) xx .^(5)C_(3) - .^(5)C_(3) xx 2^(3) xx .^(5)C_(2) + .^(5)C_(4) xx 2^(4) xx .^(5)C_(1) - .^(5)C_(5) xx 2^(5) xx .^(5)C_(0)`
`= 1-5 xx 5 xx 2 + 10 xx 10 xx 4 - 10 xx 10 xx 8+ 5 xx 5 xx 16 - 32`
`= -81`
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