r balls in n urns

Yankel

Active member
Hello all,

I am trying to solve this problem:

r balls are randomly assigned into n urns. The assignment is random and the balls are cannot be distinguished. What is the probability that exactly m urns will contain exactly k balls each ?

I know that the probability of each ball to be in each urn is 1/n.
I addition I have (nCm) ways to choose which are the urns to be filled with k balls.
Then I have r−km balls to distribute in the n−m remaining urns.

I do not know how to proceed. Can anyone solve this difficult problem ?

My current attempt is:

$\frac{\binom{n}{m}\binom{r-km+n-m-1}{r-km}}{\binom{r+n-1}{r}}$

It must be wrong....

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