Assume we have a set of $n$ balls. For each step, we uniformly pick one ball and label it if it is not labeled. Or otherwise move on to next step. I am wondering what is the distribution of number of steps till $m$ balls are labeled in the set.

This r.v. could be expressed as $G_1+G_2+\dots+G_m$, where $G_k$ is a geometric distribution with parameter $\frac{(n-k+1)}n$. It's conditional on $(k-1)$ balls have been labeled and how many step we need to reach one of those unlabeled balls.

Do we have a specific name for this r.v.? I am curious about its concentration around the mean and if it is unimodal discrete distribution.

Intuitively it is like a generalization of negative binomial distribution.