I'm dealing with a physics problem right now that essentially boils down to a ball and box problem. I know that the distribution of N indistinguishable balls into k distinguishable boxes is given by the multi-nomial distribution.
The problem that I'm working is a variant, where there are now two distinct sets of distinguishable boxes, and the number of balls distributed into each set can vary (i.e. $N$ = $N_{1}$ + $N_{2}$ where $N$ is fixed but $N_1$ and $N_2$ can vary.) I'm not sure what distribution describes this situation?