Question

We go to a pizza party, and there are 5 types of pizza. We havestarved for days, so we can eat 13 slices, but we want to sample eachtype at least once. In how many ways can we do this? Order doesnot matter.

Answer 1

First, you eat the 5 types each, which only leaves for 8 more. So the top of the would be (8+5-1), for the choose part that means there 1 less type of pizza to choose from which makes the equation

$\binom{8+5-1}{5-1}=\binom{12}{4}$$\binom{12}{4}=\frac{12!}{4!(12-4)!}=\frac{12\cdot10\cdot11\cdot10\cdot9}{4\cdot3\cdot2\cdot1}=495$

This means that there are 495 ways you can do this.