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:)

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put on hold as off-topic by David White, RP_, user44191, Neil Hoffman, LSpice May 15 at 19:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – David White, RP_, user44191, Neil Hoffman
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ MSE is a right forum for such type questions. As a first step, consider it for concrete values of parameters. $\endgroup$ – user64494 May 14 at 18:38

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