The original coupon collector problem was answered here:
Probability distribution in the coupon collector's problem
The subset version of the coupon collector's problem is:
There are $m$ different kinds of coupons to be collected from boxes. Assuming each type of coupon is equally likely to be found per box, what's the expected amount of boxes $(N)$ one has to buy to collect $q$ types of coupons, where $q<m$?
My question is, what is the "closed form" (as a sum perhaps) of $P(N\leqslant n)$?
Note that the collector is seeking a specific given subset of the $m$ coupons, rather than any $q$ of them.
Best Answer
You can solve this exactly like the original coupon collector's problem. At the beginning, the probability that a randomly-selected coupon is one of the desired ones is $\frac qn$, so the expected waiting time is $\frac nq$. The expected waiting time for the second coupon is $\frac n{q-1}$ and so on, down to the last coupon, which has an expected waiting time of $n$. The overall expected waiting time is $$\sum_{k=1}^q\frac nk = nH_q\approx n\log q$$