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When applying Polya's counting theorem one needs to find the coefficients of complicated polynomials. Example:

Find the coefficient of $r^2w^3b^4$ in $4(r+b+w)^3(b^2+r^2+w^2)^3$.

In this case I know the answer is 120 but I am not sure how to get there. I have seen several examples about finding coefficients when only when factor is present but I am unsure how to proceed when several polynomials are multiplied together. Any hints would be much appreciated. I am aware of the binomial and multinomial theorems but I'm having trouble invoking them in this case.

(Apologize for the previous version of this question that was posted and quickly deleted. I thought I made a trivial mistake but then realized I had not. This one will stay up. )

kjetil b halvorsen
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1 Answers1

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Not a general approach but for this case you can do with simple arithmetic without expanding the groups.

Let degree is the sum of exponents

First, drop the coeff 4. First term contributes 3 and the second term 6. Note also that second term can only contribute even powers.

Now the grouping can only be $(w^3, r^2b^4), (r^2w, w^2b^4), (wb^2,r^2w^2b^2)$

Using the binom coefficients for degree 3. Sum of coefficients will be $1\cdot 3+3 \cdot 3+3\cdot6 = 30$

karakfa
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