Two friends and a 3 slice pizza - what can happen? Well, we'd guess they'd each eat a slice and fight over the last one. But, allowing for the vagaries of appetite and politeness, there might be a slice left over. Maybe they aren't even hungry at all, so all the pieces are uneaten. How many different possibilities are there? If the friends are named A and B, and we make up a third invisible friend named C who gets the uneaten slices, then the possibilities are:
| A | B | C |
|---|---|---|
| 3 | 0 | 0 |
| 2 | 1 | 0 |
| 2 | 0 | 1 |
| 1 | 2 | 0 |
| 1 | 1 | 1 |
| 1 | 0 | 2 |
| 0 | 3 | 0 |
| 0 | 2 | 1 |
| 0 | 1 | 2 |
| 0 | 0 | 3 |
Thus, there are 10 ways in which the (distinguishable) friends can divide the (indistinguishable) slices, allowing for leftovers.
Using what you have learned, consider what happens when four friends sit down to enjoy a pizza that has been cut into 12 identical slices. In how many ways can the pizza be shared, with the possibility that some (or all) of the slices will be uneaten?
I give up, show me the solution.