The [coupon collector's problem][1] is a problem in probability theory that states the following (from wikipedia):

> Suppose that there are $n$ coupons, from which coupons are being collected with replacement. What is the probability that more than $t$ sample trials are needed to collect all $n$ coupons?

A generalization of this problem was proposed by Newmann & Shepp, by requiring that $k$ samples of each coupon be collected. The answer to this is known. 

I, however, need to calculate the answer to an even further generalization, which is:

> How many sample trials are needed to collect a certain subset of $n$, call it $m$, at least $k$ times?

Any help or a point in the right direction would be greatly appreciated.

  [1]: http://en.wikipedia.org/wiki/Coupon_collector's_problem