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## Homework Statement

Find the coefficient of the [tex] x^{12}y^{24} [/tex] for [tex] (x^3+2xy^2+y+3)^{18} [/tex].

## Homework Equations

Multinomial theorem, as stated on http://en.wikipedia.org/wiki/Multinomial_theorem

## The Attempt at a Solution

Using the multinomial theorem in the form of the wikipedia post, I would set [tex] x_1=x^3, x_2=2xy^2, x_3=y, x_4=3 [/tex]. Now I need to find the k's that "match" the coefficients of the term from the problem statement. This will give me a relation between the k's of the following form

[tex] 3k_1+k_2 = 12, 2k_2+k_3 = 24, and k_1+k_2+k_3+k_4=18 [/tex]. Now I let [tex] k_1 [/tex] vary and record the discrete values [tex] k_2, k_3, k_4 [/tex], but what I'm confused on is why do I sometimes get negative values for [tex] k_4 [/tex]? Do I need to do something different with the relationship of the k's. Thanks in advance.