In an example in my textbook, I came across a question where it was asked to find the generating function for the number of partitions of ${n \in N}$ into summands that (a) cannot occur more than 5 times; and (b) cannot exceed 12 and cannot occur more than 5 times.
The proposed solution for (a) is:
${f(x)=(1+x+x^2+...+x^5)(1+x^2+x^4+...+x^{10})...=\Pi_{i=1}^\infty(1+x^i+x^{2i}+...+x^{5i})}$
And the solution for (b) is proposed as:
${\Pi_{i=1}^{12}(1+x^i+x^{2i}+...+x^{5i})}$
However, the book fails to elaborate on what these problems concretely represent in practical terms.
For instance, when it says, it cannot occur more than 5 times, would it mean that we cannot repeat the same part of the partition more than 5 times? And what exactly does it mean when it says, it cannot exceed 12 in practical terms?
Think of playing a solitaire game where you have $5$ identical decks of $m$ cards; the cards are numbered $1,\ldots,m$. You mix the decks together and try to find sets adding up to $n$. The question is how many solutions there are.
For (a), you can think of $m$ as being $\infty$, i.e. there's no bound on the numbers on the cards. Or you can take $m=n$, since cards with labels greater than $n$ can't be in a solution.
For (b) we are fixing $m=12$.