Since the items are indistinguishable and the baskets are indistinguishable, we are just looking for ways to split up the number 5 into 3 or fewer numbers.
The way I made sure to not miss any was to start with the partitions that started with 5, then 4, then... to 2, and put the numbers in descending order.
They are 5, 4+1, 3+2, 3+1+1, 2+2+1. No partitions can start with 0 or 1, as then it is impossible to reach 5 by adding the numbers up. It is easy to see that all partitions have been listed, since there is only 1 way to sum 1, two ways to sum 2, and so on.
This makes a total of 5 different ways.
