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+...+G_m, where G_k is a geometric distribution with parameter (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.