How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable but the boxes are?

Since the balls are indistinguishable, we must only count the number of balls in the different boxes.

There are  ways to arrange the balls as  (specifically, box 1 can have 5, box 2 can have 5, box 3 can have 5).

There are  to arrange  and  ways to arrange ; in each case, we must choose one of the 3 boxes to have the largest number of balls, and also one of the remaining two boxes to be left empty.

However, there are only  ways to arrange , and  ways to arrange ; in each case, we must choose one box to have the `different' number of balls (3 in the  case and 1 in the  case).

This gives a total of  arrangements.