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

Answer 1

There are many ways to approach this problem, but one way to think about it is to use a combination of distributions with exactly one box empty and distributions with more than one box empty.

If one box must be empty, there are 5 ways to choose which box will be empty. Then, we need to distribute the remaining 8 balls among the 4 remaining boxes. This can be done using stars and bars, which is a method used to count the number of ways to distribute indistinguishable objects into distinguishable boxes.

Using stars and bars, the number of ways to distribute 8 balls among 4 boxes is (8+4-1 choose 4-1) = 11 choose 3 = 165.

Therefore, the total number of ways to distribute 8 balls among 5 boxes, with at least one box empty, is 5 * 165 = 825 ways.