Question

I have four identical oranges. How many ways are there for me to divide these oranges into at most three groups? (By definition, a group must have at least one orange.)

Answer 1

Group 1 - all groups of 1 orange. There are only 4 oranges, so 4 groups
Group 2 - all groups of 2 oranges. 4 oranges arranged in groups of 2 is 4 choose 2
Group 3 - all groups of 3 oranges. 4 oranges arranged in groups of 3 is the same as 4 oranges arranged in groups of 1
Add all these 3 possibilities.