The sandwich shop offers combo meals with a choice of one sandwhich, one side, and one drink. In total, there are 560 combos available. if there are 16 sandwiches and 5 drinks available, how many sides are there?

This question is a slight variation of the following: If there are 16 ways to pick a sandwich (16 choices), x ways to pick a side and 5 ways to pick a drink, how many lunches are there consisting of exactly one sandwich, one side and one drink?

In these sorts of problems, where each choice is independent of the other (you can pick whatever drink you want regardless of the side or sandwich - they do not depend on each other) you get the number of lunches by multiplying all of the ways each things can be picked. So here we would do:

(# of sandwiches) (number of sides) (number of drinks) = number of lunches

Let’s fill in what we know:

(16) (number of sides) (5) = 560

80x = 560

I made x = number of sides.

So we divide both sides by 80 and obtain x = 7.

There are 7 choices for sides.

The property that allows for multiplication in these types of problems is called the counting principle.