# Certain urn models [on hold]

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:)

New contributor
daniel is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.

## put on hold as off-topic by David White, RP_, user44191, Neil Hoffman, LSpiceMay 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.

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