Will every room get a guest?

Question

A small hotel has 5 rooms. Seven guests each independently choose a room uniformly at random. What is the probability that every one of the 5 rooms ends up with at least one guest?

Accepted Answer

There are 5^7 = 78125 equally likely ways to assign the seven guests to rooms. We need to count the assignments that leave no room empty, then divide. Counting "no room empty" directly is hard, so use inclusion-exclusion: start from all assignments, subtract those that miss at least one specific room, add back those that miss at least two (subtracted twice), and so on. If a set of k particular rooms is banned, the guests must fit into the other 5 - k rooms, giving (5-k)^7 assignments, and there are \binom{5}{k} ways to pick which k rooms are missed. So the count of assignments hitting all roo…

Will every room get a guest?

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