Find the coefficient of `t^8` in the expansion of `(1+2t^2-t^3)^9`.

Options

A. `1680`
B. `2140`
C. `2520`
D. `2730`

Answer

C

Explanation

`(c )` `((1+2t^(2))-t^(3))^(9)`
`="^(9)C_(0)(1+2t^(2))^(9)-^(9)C_(1)(1+2t^(2))^(8)*t^(3)+^(9)C_(2)(1+2t^(2))^(7)*t^(6)-^(9)C_(3)(1+2t^(2))^(6)*t^(9)+.....-^(9)C_(9)(t^(3))^(9)`
`:.` Coefficient of `t^(8)` in the expansion of `(1+2t^(2)-t^(3))^(9)`
`=^(9)C_(0)("coefficient of" t^(8) "in" (1+2t^(2))^(9)) -^(9)C_(1)("coefficient of" t^(5) "in" (1+2t^(2))^(8))+^(9)C_(2)("coefficient of"t^(2) "in" (1+2t^(2))^(7))`
`=^(9)C_(9)*^(9)C_(4)2^(4)-0+^(9)C_(2)*^(7)C_(1)*2`
`=2520`