# Multinomial Theorem Problem

1. Jan 20, 2012

### tylerc1991

1. The problem statement, all variables and given/known data

What is the coefficient on $x^{12} y^{24}$ in $(x^3 + 2xy^2 + y + 3)^{18}$?

2. Relevant equations

Multinomial Theorem:
$\displaystyle \left( \sum_{k = 1}^m x_k \right)^n = \sum_{k_1 + \dotsb + k_m = n} \binom{n}{k_1, \dotsc, k_m} \prod_{i = 1}^m x_i^{k_i}$

3. The attempt at a solution

Using the multinomial theorem, the expansion of $(x^3 + 2xy^2 + y + 3)^{18}$ has terms of the form

$\binom{18}{k_1, k_2, k_3, k_4} (x^3)^{k_1} (2xy^2)^{k_2} (y)^{k_3} (3)^{k_4} = \binom{18}{k_1, k_2, k_3, k_4} x^{3k_1} 2^{k_2} x^{k_2} y^{2k_2} y^{k_3} 3^{k_4} = \binom{18}{k_1, k_2, k_3, k_4} x^{3k_1 + k_2} 2^{k_2} y^{2k_2 + k_3} 3^{k_4}.$

The $x^{12} y^{24}$ occurs when $3k_1 + k_2 = 12$ and $2k_2 + k_3 = 24$. We also know that $k_1 + k_2 + k_3 + k_4 = 18$. This is where I get stuck, since I have 3 equations and 4 unknowns. Is my work correct up to this point? Is there another equation that I can use to find each $k_i$? Thank you!