The coefficient of `x^(10)` in the expansion of `(1+x^2-x^3)^8` is `476` b. `496` c. `506` d. `528`

Options

A. 420
B. 476
C. 532
D. 588

Answer

b

Explanation

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`