What kind of distribution answers this question?

Suppose that I have $N$ distinct objects, each of which have $1/N$ chance of being drawn. What is the expected number of draws, $D$, to draw at least $x$ of each object (with replacement)? For example: there are 30 object. What is the expected number of draws I would need to get at least 1 of each object?

For the record, this isn't a homework question or anything, but just something I was thinking about randomly as I was playing a game. What kind of distribution is the above?

• The expected number of draws to get at least once each object is $\frac1N+\frac1{N-1}+\cdots+\frac12+1$, as shown here. The distribution of the number of draws can be deduced from the same approach.
– Did
Mar 14 '15 at 13:22
• This is an example of the coupon collecting problem (assuming all coupons are equally likely). Mar 15 '15 at 23:53