There are $n$ buckets, each with a possibly different finite capacity $c_i \; (i=1, \dots, n)$. There are $k$ balls, each to be distributed randomly to the buckets. A bucket cannot be filled over its capacity. I would like to know the pmf of the number of balls that bucket $i$ receives.
I saw this post, Probability of assigning balls into buckets, where each bucket has a certain capacity., but it doesn't quite capture what I want. I thought about just using a Markov chain, but there could potentially be too many states in the probability transition matrix for my problem. It would grow exponentially fast.
Is there a formula for the pmf of the number of balls that bucket $i$ receives?
Edit: If it was not clear, initially buckets are chosen uniformly at random, and when a bucket reaches its capacity, it drops out and buckets are chosen uniformly at random from those buckets not yet filled.