TITLE: How to solve a generalization of the Coupon Collector's problem
QUESTION [9 upvotes]: The coupon collector's problem 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 at least $k$ coupons of $m$ different types?

Any help or a point in the right direction would be greatly appreciated.
(Edit#3): removed the misleading example.
(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?

REPLY [3 votes]: This is given in Thm 3.1 of http://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf