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

Question 1207625: In how many ways can you distribute $8$ indistinguishable balls among $5$ distinguishable boxes, if at least three of the boxes must be empty?
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

3 empty boxes....

The 8 balls must be distributed to 2 boxes. The number of ways of choosing 2 of the 5 boxes is C(5,2) = 10. Using stars and bars (which I will assume you are familiar with, since you are working this kind of problem), the number of ways to distribute the 8 balls in 2 boxes is C(9,1) = 9.

Number of ways to distribute the balls if 3 boxes are empty: 10*9 = 90

4 empty boxes....

The 8 balls must be "distributed" to a single box. There are 5 boxes to choose from; and for each box chosen there is only one way to put the 8 balls in that box.

Number of ways to distribute the balls if 4 boxes are empty: 5*1 = 1

ANSWER: 90+5 = 95

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