Problem:

Find a generating function for the number of ways to distribute n identical objects to five identical boxes.


I already looked in the back of the book and saw that the answer is (1 + x + x^2 + ...)(1 + x^2 + x^4 + ...)...(1 + x^5 + x^10 + ...) but I can't understand how they got the answer.

I'm assuming the parts implied by the third ellipses are (1 + x^3 + x^6 + ...)(1 + x^4 + x^8 + ...), and I see some kind of sense to it, but going off what I know about generating functions, I can't understand why for each box only multiples of 2, multiples of 3, etc. can be placed in certain boxes.