When we allocate $N:=k\cdot Z$ balls, where $\frac{N}{2}$ are red balls and $\frac{N}{2}$ are black balls, into $k$ groups, the number of red balls in each group follows a multivariate hypergeomtric distribution. Then, how to estimate or get a tight upper bound on its entropy?
I could get some trivial uppder bounds as follows: 1. using the expectation to estimate, 2. esitimate by $\sum_{x\in\mathcal{X}}\max -p_x \log p_x$. Both results seem meaningless for me.