How many ways are there to put 5 balls in 3 boxes if the balls are not distinguishable but the boxes are?
There are 10 ways to put 5 balls in 3 boxes. You can either have all the balls in one box, all but one ball in one box, or one ball in each box. Here are the combinations:
5 balls in one box, 0 in the other two
4 balls in one box, 1 in another box, 0 in the last box
4 balls in one box, 0 in another box, 1 in the last box
3 balls in one box, 2 in another box, 0 in the last box
3 balls in one box, 1 in another box, 1 in the last box
3 balls in one box, 0 in another box, 2 in the last box
2 balls in one box, 2 in another box, 1 in the last box
2 balls in one box, 1 in another box, 2 in the last box
1 ball in one box, 2 in another box, 2 in the last box
0 balls in one box, 3 in another box, 2 in the last box
This gives us a total of 5 + 4 + 4 + 3 + 3 + 3 + 2 + 2 + 1 = 27 ways.
