Suppose there are $N$ objects, of which $K$ are of type 1 and $N-K$ of type 2. Objects of type 1 are indistinguishable and objects of type 2 are indistinguishable. One is interested in the probability of getting $k$ objects of type 1 in $n$ draws from $N$, with replacement. This probability then is given by
$$\frac{\frac{K^{k}}{k!} \frac{(N-K)^{n-k}}{(n-k)!}}{\frac{N^{n}}{n!}}$$
Does this probability distribution have a name? It looks like it's related to hypergeometric distribution with the difference being that here we are drawing with replacement...
Edit. Taking another look at it, one can write the above as
$$\frac{n!}{k!(n-k)!}\frac{K^{k}}{N^{k}}\frac{(N-K)^{n-k}}{N^{n-k}}$$ which is the binomial distribution with parameter $p = \frac{K}{N}$, i.e. the probability of "success" when you perform $n=1$ draws, right?