In the standard Coupon Collector problem, someone is choosing coupons at random (with replacement) from n different possible coupons. Duplicate coupons do us no good; we need a complete set. The standard question is "What is the expected number of coupons (or probability distribution in number of coupons) to collect them all?
In the standard birthday problem, we choose k items from n options with replacement (such as k people in a room, each with one of 365 possible birthdays) and try to determine the probability distribution for how many unique values there will be (will they have the same birthday?).
In my problem, someone has chosen k items from n options and I know that there were p distinct values, but I don't know what k was. If p=n this is the coupon problem, but I want to allow for values of p that are less than n. I want to determine the probability distribution for k (actually, all I need is the expected value of k, but the distribution would be interesting as well) as a function of p and n.