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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 8, 2024
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There are two cases to consider:

 

Case 1: One box is empty


There are 5 ways to choose which box will be empty. The remaining 7 balls can be distributed among the remaining 4 boxes using stars and bars formula. Therefore, the total number of ways in this case is 5 * (7+4-1) choose (4-1) = 5 * 10 = 50 ways.

Case 2: Two boxes are empty


There are a total of 5 choose 2 = 10 ways to choose which 2 boxes will be empty. The remaining 6 balls can be distributed among the remaining 3 boxes using stars and bars formula. Therefore, the total number of ways in this case is 10 * (6+3-1) choose (3-1) = 10 * 8 = 80 ways.

Adding the two cases together, the total number of ways to distribute 8 indistinguishable balls among 5 distinguishable boxes if at least one of the boxes must be empty is 50 + 80 = 130 ways.

 Jun 9, 2024

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