Question for Counting using PIE

Question

After another gym class you are tasked with putting the 13 identical dodgeballs away into 3 bins. Each bin can hold at most 5 balls. How many ways can you clean up?

Answer 1

Let us label the bins as Bin 1, Bin 2, and Bin 3. Let (i,j,k) denote the outcome where i balls are in Bin 1, j in Bin 2, and k in Bin 3. We must have the i+j+k = 13. and each of i, j, and k are 3, 4 or 5 (Note that no bin can have less than 3 balls, as then the other two bins would have 11 or more balls, which is not possible, as each bin can hold no more than 5 balls.)

The different possibilities are (5, 5, 3), (5, 3, 5), (3, 5, 5), (5, 4,4), (4, 5,4) and (4,4,5).