[math-fun] coins in barrels

For a change of pace, here's a real probability question: Cheng Qian, an engineering student doing work on visual sensor networks, asks:

I throw n coins into m barrels randomly. (For a single coin, each barrel has same chance to get it.) Then I count the number of coins in each barrel, say k(i), i=1...m .

My question: what is the probability that all the barrels have no more than K coins, i.e. k(i)<=K, for i=1..m.

Does anyone know any good off-the-shelf approximations to this? (I assume that an exact formula would be too much to ask for.) This isn't exactly "fun" math, but it's a simple and natural enough question that I thought it wouldn't be out of place in this group. Jim Propp