The coefficient of `x^5` in the expansion of `(x^2-x-2)^5` is `-83` b. `-82` c. `-86` d. `-81`

Options

A. -83
B. -82
C. -81
D. 0

Answer

C

Explanation

We have,
` = {x^(2) - (x + 2)}^(5)`
= `""^(5)C_(0) (x^(2))^(5) - ""^(5)C_(1) (x ^(2))^(4) (x +2) + ""^(5)C_(2) (x^(2))^(3) (x + 2)^(2)`
` - ""^(5)C_(0) (x^(2))^(2) (x + 2)^(3) + ""^(5)C_(4) (x ^(2)) (x +2)^(4) - ""^(5)C_(5) (x + 2)^(5)`
`therefore ` Coefficients og `x^(5)` in `(x^(2) - x - 2)^(5)`
` = - ""^(5)C_(3) xx` coefficients of x in `(x + 2)^(3)`
`+""^(5)C_(4)xx` Coefficient of `x^(3)` in `(x + 2)^(4)`
`-""^(5)C_(5) xx` Coefficient of `x^(5)` in `(x + 2)^(5)`
` = - ""^(5)C_(3)xx""^(3)C_(2)xxC^(2)+""^(5)C_(4)xx""^(4)C_(1)xx2 - ""^(5)C_(5)xx1`
`= - 120 + 40 - 1 = -81`