# Combinations of a Multi-set with generating functions

Find the number of 5-combinations of the multi-set {4*a, 4*b, 4*c} using generating functions.

Because I am finding combinations I know that I have to use ordinary generating functions and not exponential generating functions.

I can first represent this as $$(1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4)$$ $$=(1+x+x^2+x^3+x^4)^3$$

I'm not sure where to go from here though- I would greatly appreciate any advice or help! Thanks.

• You could expand that polynomial (though that's probably not the cleanest way to do it) – Dennis Meng Dec 5 '13 at 18:32
• After expanding it, what would I do to find the exact number of 5-combinations? – user2553807 Dec 5 '13 at 19:42
• As already mentioned in the answer, you'd be looking for the coefficient to $x^5$. (Remember, the exponents in generating functions mean something; it's not just a magic formula) – Dennis Meng Dec 5 '13 at 19:45
• If I'm looking at the coefficient of x^5 wouldn't it be -3? Can you explain your comment a little more or help me find more information on that. Thanks!! – user2553807 Dec 5 '13 at 21:44
• Given that the polynomial has zero negative signs, I haven't a clue where you got a negative sign from. – Dennis Meng Dec 5 '13 at 21:48

This is the coefficient of $x^5$ in the expansion of $(1+x+x^2+x^3+x^4)^3 = (1-x^5)^3/(1-x)^3 = (1-3x^5 +3x^{10} -x^{15})\displaystyle\frac{1}{(1-x)^3}$. Can you find a taylor series expansion for the second term in the product?