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?