Extraction of coefficient from Generating Function with partitions

Question

Determine the coefficient of $~x ^ {15}~$ in:

$(1+^3+^6+^9+^{12}+^{15})(1+^6+^{12})(1+^9)$

How to use the fact that the desired coefficient is the number of partitions of 15 in parts restricted to the set {3,6,9} to compute it?

Answer 1

You can multiply out the second and third terms to get $$(1+x^3+x^6+x^9+x^{12}+x^{15})(1+x^6+x^9+x^{12}+x^{15}+x^{21})$$

Then observe that each subterm in wlog the first term pairs up with at most one subterm in the other term...