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The coefficient of x^(10) in the expani...

The coefficient of ` x^(10)` in the expanion of
` (1 + x^(2) - x^(3))^(8)` , is

A

420

B

476

C

532

D

588

Text Solution

Verified by Experts

The correct Answer is:
b
Given expansion can be rewritten as `[1 + x^(2)(1-x)]^(8)`
`= ""^(8)C_(0) + ""^(8)C_(1) x^(2)(1-x)+ ""^(8)C_(2) x^(4) (1-x)^(2)`
` + ""^(8)C_(3) x^(6) (1-x)^(3) + ""^(8)C_(4) x^(8) (1-x)^(4) + ""^(8)C_(5) x^(10) (1-x)^(5)+…`
There are only two terms , which we get the coefficient of ` x^(10). `
` therefore ` Coefficient of ` x^(10) = ""^(8)C_(4) ` [Coefficient of ` x^(2) "in" (1-x)^(4)]`
` + ""^(8)C_(5)` [Coefficient of ` x^(6) "in" (1-x)^(5)]`
` = ""^(8)C_(4) (""^(4)C_(2)) + ""^(8)C_(5) (1)`
` = (""^(8)C_(4)) (""^(4)C_(2)) + ""^(8)C_(3) = (70)(6) + 56= 476`
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