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

\(Edit:\) Here is an example to illustrate the problem:

> Let's say we would like to collect
> coupons that come with a certain brand
> of cereal. We know there are 10 (read:
> $n$) different types of coupons , but
> we are really only interested in 5
> (read: $m$) of the 10 (since these
> coupons apply to products we would
> like to buy). We would like to also
> give sets of these 5 coupons to our
> friends, so they can share in the joy.
> So, assuming every coupon has equal
> chance to appear in any box of cereal,
> how many boxes would you need to buy
> to have 3 (read: $k$) sets of the 5
> coupons you are interested in?

The difference that $m$ introduces to the problem is the following: For every coupon you collect, the odds of collecting a **new** coupon becomes less. If it was only $n$, then when you have collected all but the last coupon your chance of getting the last one in the next trial is $1/n$. If you are only interested in collecting $m$ though, the chance of collecting the last coupon on the next trial is $(n-m+1)/n$

**(Edit #2)**: This problem can be stated simply as a balls-and-bins problem. If we have $n$ bins, how many balls need to be thrown so that at least $m$ bins have at least $k$ balls?