I recently encountered the following interesting problem:
Suppose that I put 6 identical dice in a cup, and roll them simultaneously (as in Farkle, for example). Then you take those same 6 dice, and roll them all again. What is the probability that we both observe the same outcome? For example, we may both roll one of each possible value (1-2-3-4-5-6, but not necessarily in order), or we may both roll three 3s and three 6s, etc.
I like this problem as an “extra” for combinatorics students learning about generating functions. A numeric solution likely requires some programming (but I’ve been wrong about that here before), but implementation is not overly complex, while being slightly beyond the “usual” type of homework problem in the construction of its solution.