Five balls into three boxes: no box left empty

Question

Five distinguishable balls are each dropped independently and uniformly at random into one of three distinguishable boxes. What is the probability that every box ends up with at least one ball?

Accepted Answer

Each ball has 3 choices, so there are 3^5 = 243 equally likely outcomes. We want the outcomes where no box is empty. Count the "bad" outcomes (at least one empty box) with inclusion-exclusion. Let Ai be the event that box i is empty. If a specific box is empty, the five balls use the other two boxes: 2^5 = 32 ways, and there are \binom{3}{1}=3 boxes. If two specific boxes are empty, all balls go into one box: 1^5 = 1 way, with \binom{3}{2}=3 choices. Three boxes cannot all be empty. So \(\text{some box empty}) = 3 \cdot 2^5 - 3 \cdot 1^5 = 96 - 3 = 93. The favorable count (no empty box) is 24…

Five balls into three boxes: no box left empty

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