Suppose you are collecting coupons where each of the n types of coupons have a uniform probability of occurring when a purchase is made. To win one requires k coupons of the same type (can be any type). I'm trying to find a reoccurrence relation that will give the expected number of purchases that need to be made before one gets their kth coupon of any type.
I've been stuck on this problem for awhile and am looking for any hints on how to proceed. I know that this is a variation of the standard coupon problem, but I haven't been able to figure out a means of finding the expected number of purchases based on k and n, or any examples of this kind of coupon problem.