Suppose that you made a webpage and you are collecting the statistics from the visitors. There are m types of visitors (students, company recruiters, etc.). Each visit is equally likely to be any of the m types, independent of the types of previous visitors. Find the expected number of visitors needed in order to have at least one of each type.

This is a homework question.
As far as I know, let just say the number of each visitor is X_i, and it is geometrically distributed with p=1/m
Let's call the number of visitor needed X: X = sum(X_i)
The problem is, what is the distribution of X.