Consider an urn containing $N$ balls of $P$ different colors, with $p_i$ being the proportion of balls of color $i$ among the $N$ balls ($\sum_i p_i = 1$). I draw $n \leq N$ balls from the urn without replacement and look at the number $\gamma$ of different colors among the balls that were drawn. What is the expectation of $\gamma$ as a function of $n/N$, depending on suitable properties of the distribution $\mathbf{p}$?
To give more insight: if $N = P$ and $p_i = 1/P$ for all $i$, then I will always see exactly $n$ colors, that is, $\gamma = P (n/N)$. Otherwise, it can be shown that the expectation of $\gamma$ is $>P(n/N)$. For fixed $P$ and $N$, it would seem that the factor by which to multiply $n/N$ would be maximal when $\mathbf{p}$ is uniform; maybe the expected number of different colors seen be bounded as a function of $n/N$ and, e.g., the entropy of $\mathbf{p}$?
This seems related to the coupon collector's problem, except that sampling is performed without replacement, and the distribution of the coupons is not uniform.