The card game Doppelkopf is played with four players. Every player receives 12 of the 48 cards. The 48 cards consist of 26 trump cards and 22 other. A trump poverty is what we call the scenario that a player has less than 4 trump cards. If the cards are dealt randomly, what is the probability that at least one player has a trump poverty?
This looks like it can be solved very easily via the inverse, like so:
Deal every player 4 trump cards, then deal every player 8 more cards.
However we failed to do this without double counting. Can this be done without case distinctions?