In how many ways can you distribute $8$ indistinguishable balls among $5$ distinguishable boxes, if at least three of the boxes must be empty?
2 boxes with balls in them
Choose 2 boxes of the 5 = C(5,2) = 10 ways of selecting the boxes
Ball splits Ways to distribute the balls
7 1 2
6 2 2
5 3 2
4 4 1
2 * 2 * 2 * 1 = 8 ways
So 10 * 8 = 80 ways
1 box with all 8 balls = 5 ways to do this
Total ways
80 + 5 = 85
+2653 
