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In how many ways can you distribute 8 indistinguishable balls among 5 distinguishable boxes, if at least one of the boxes must be empty?

 Jun 29, 2023
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There are two cases to consider:

Case 1: One of the boxes is empty. In this case, there are 4 ways to choose which box is empty, and then 7 ways to distribute the remaining 8 balls among the 4 boxes. So there are 4×7=28 ways in this case.

Case 2: No boxes are empty. In this case, there are 3 ways to choose which box has 1 ball, 2 ways to choose which box has 2 balls, and 1 way to choose which box has 3 balls. So there are 3×2×1=6 ways in this case.

Therefore, the total number of ways to distribute the balls is 28+6=34​.

 Jun 29, 2023

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