Say I have K balls and S urns. If I randomly distribute each of the K balls to urns (independently), what is the expected value of the product of the amount of balls in each urn?
For example, if I had 8 balls and 3 urns, one possible distribution might be (4,2,2) and result in 4*2*2=16. (I believe the answer with K=8 and S=3 is 12.44444, but I'm not sure how to get this result).
Let $U(k,s)$ denote the indicator function of the event that ball $k$ is in urn $s$, then the (random) number of balls in urn $s$ is $$\sum_{k=1}^KU(k,s)$$ hence the product $X$ of the number of balls in each urn is $$X=\prod_{s=1}^S\sum_{k=1}^KU(k,s)$$ Since $U(k,s)U(k,t)=0$ for every $s\ne t$, expanding the product in the RHS yields $$X=\sum_{(k_1,\ldots,k_S)}\prod_{s=1}^SU(k_s,s)$$ where the sum is over every $S$-tuple of distinct integers in $\{1,\ldots,K\}$. There are $$\frac{K!}{(K-S)!}$$ such $S$-tuples and the expectation of each associated product of random variables $U(k_s,s)$ is, by independence, $$\prod_{s=1}^SE(U(k_s,s))=\left(\frac1S\right)^K$$ hence, for every $K\geqslant S$, the expected value of the product of the number of balls in each urn is $$E(X)=\frac{K!}{S^K\cdot(K-S)!}$$