This is my first question here - so if i'm out of format - let me know :)

The problem is this:

$m$ identical urns

$k$ identical cells in each urn

$n$ identical balls

The balls are distributed among the urns, with no restriction other than the maximal capacity of each urn (up to $k$ balls - one in each cell). The question at hand is: What is the probability that exactly $t$ urns are occupied by $r$ balls.

I thought of approaching it the same way Fang does (here: https://www.researchgate.net/publication/305977113_A_restricted_occupancy_problem): $P(M_t=r)=N(m,k,n,r,t)/N(m,k,n)$

Where $N(m,k,n,r,t)$ is the number of ways of distributing the $n$ balls into the $m$ urns, such that exactly $r$ of them have exactly $t$ balls.

And where $N(m,k,n)$ is the number of ways of distributing the balls into the urns.

The latter is simple to calculatr (if I'm not mistaken: $N(m,k,n)=mk(mk-1)\cdot...\cdot(mk-n+1)$), but the former seems quite hard. I have trouble using the generating functions to solve $N(m,k,n,r,t)$, because I can't find the independent variables to use (so that the generating functions can be multiplied).

I'd love it if someone could explain clearly how he would count $N(m,k,n,r,t)$. Thanks in advance:)