Probability of getting N distinct types of objects in r tries from a set of N distinct objects

Question

Given N distinct objects that have an equal chance of being drawn (with replacement), what is the probability that X draws are needed to get all N types.

Answer 1

Hint: consider the chance you get less than $N$ and subtract from $1$. You need to use the inclusion-exclusion principle. If you search the site for that you will find many examples. For $r \lt N$ it is rather easy.

Answer 2

The probability is: $$ P_X=\sum_{k=0}^{n-1}(-1)^{n-k-1}\binom{n-1}{k}\left(\frac{k}{n}\right)^{X-1}. $$