This is a simplified version of an applied probability modeling problem.
I believe I solved it, but I'm wondering if there is a more elegant approach than mine:

There are 10 boxes and 200 balls. At each trial exactly four boxes are selected
at random and exactly one ball is placed into each. Thus there are 50 trials in the game. The
boxes have equal capacity. What must that capacity be so that in 90% of the
50-trial games no box overflows?

Answer: Obviously more than 20; using binomial, 28 seems to suffice.