generating function as english statement

Question

An ordinary enumerator is given as $(1+x+x^2)^p$. This is being understood as follows:

There are 2 each of p kinds of objects.The ordinary enumerator for selecting none (or) one (or) both the objects of that kind is $(1+x+x^2)^p$

Similarly,how do i visualize (or) understand the enumerator
$(1+x^5+x^9)^{100}$ in plain english

Answer 1

Relating that polynomial to your own example, we can understand $(1+x^5+x^9)^{100}$ as follows:

There are $9$ each of $100$ kinds of objects. The ordinary enumerator for selecting none or five or all nine of the objects of that kind is $(1+x^5+x^9)^{100}$.

Here's another interpretation:

Consider three-sided dice whose faces have $0$, $5$, or $9$ pips, and roll $100$ such dice. Then the coefficient of $x^k$ in $(1+x^5+x^9)^{100}$ is the number of ways to get a total of $k$.