Question

Coefficient of X^19(2x^3 - 3x)^9

Answer 1

By the binomial theorem,

`(2x^3-3x)^9=\displaystyle\sum_(k=0)^9\binom9k(2x^3)^(9-k)(-3x)^k=2^9\sum_(k=0)^9\binom9k \left(-\frac32\right)^k x^(27-2k)`

where

`\dbinom nk=(n!)/(k!(n-k)!)`

is the binomial coefficient.

The x¹⁹ terms occurs for 27 - 2k = 19, or k = 4, which has coefficient

`2^9\dbinom94\left(-\frac32\right)^4=\boxed{326,592}`