This is a sister question to this question. We want to place N balls of $C$ different colors into K boxes, where the boxes must contain exactly $k$ balls. The number of balls of any specific color is $b_c$, and there are $B=k*K=\sum_c b_c$ balls. Balls of the same color are not distinguishable. No 2 or more balls of the same color are allowed to be in the same box, so obviously also $b_c<K \forall c$, so each box contains either zero or one ball of any specific color. All allowed distributions of balls are supposed to be equally likely, in the sense that the probability of x co-occurrences is directly proportional to the number of configurations that have that number of co-occurrences.
Is there an explicit probability distribution for the number of co-occurrences of balls of a set of colors?
Example: We have $9$ balls of $5$ colors and $3$ boxes of size $3$. The respective numbers of balls are: $(b_1, b_2, b_3, b_4, b_5) = (2,2,2,1,2)$. Is there an explicit probability distribution of how likely it is that balls of colors $(1,2,3)$ will appear in the same box $0$ times, $1$ times or $2$ times? Any help is greatly appreciated!
The same problem with boxes of unrestricted size and balls of only $2$ colors results in a hypergeometric probability distribution, see HERE. But this problem is different enough so that it should have a different distribution.
