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The cofficient of x^(18) in the product ...

The cofficient of `x^(18)` in the product `(1 + x) (1 - x)^(10) (1 + x + x^(2))^(9)` is

A

84

B

-126

C

-84

D

126

Text Solution

Verified by Experts

The correct Answer is:
A
Given expression is
`(1 + x) (1 - x)^(10) (1 + x + x^(2))^(9)`
`= (1 + x) (1 - x) [(1 - x) (1 + x + x^(2))]^(9)`
`= (1 - x^(2)) (1 - x^(3))^(9)`
Now, coefficient of `x^(18)` in the product
`(1 + x) (1 - x)^(10) (1 + x + x^(2))^(9)`
= coefficient of `x^(18)` in the product `(1 - x^(2)) (1 - x^(3))^(9)`
= coefficient of `x^(18)` in `(1 - x^(3))^(9)`
- coefficient of `x^(16)` in `(1 - x^(3))^(9)`
Since, `(r + 1)^(th)` term in the expansion of
`(1 - x^(3))^(9)` is `.^(9)C_(r ) (-x^(3))^(r ) = .^(9)C_(r ) (-1)^(r ) x^(3 r)`
Now, for `X^(18), 3r = 18 implies r = 6`
and for `x^(16), 3r = 16`
`implies r = (16)/(3) cancel(in) N`.
`:.` Required coefficient is `.^(9)C_(6) = (9!)/(6! 3!) = (9 xx 8 xx 7)/(3 xx 2) = 84`
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