# What is the name of this distribution?

I came across this: a categorical distribution with $$K=10,000$$ parameters (categories), and we take only few samples from this distribution, say $$N=400$$ (the point is $$N < K$$). Now, obviously, not all categories will show up in the sampled data, so let's call number of categories observed $$x$$, $$x$$ is a random variable, what do you call the distribution of $$x$$?

Let $$X_1, \dotsc, X_N \sim \mathcal{Catdist}(K,\pi_1,\dotsc,\pi_K)$$ (and interest is in case $$N). Let $$Y_j=\sum_i \{X_i=j\}$$ be the count of the number of times the value $$j$$ was observed. Now $$(Y_1, \dotsc,Y_K)\sim\mathcal{multinom}(N,\pi_1,\dotsc,\pi_K)$$.
Your question is about the random variable $$B=\sum_{j=1}^K \{Y_j \ge 1\}$$, I doubt there is a conventional name for this distribution. But in the special case that all $$\pi_i=1/K$$, this general problem is known as balls in boxes and something might be known. Have a look at this math SE post. There is also some similarity to occupancy problem in ecology.