# The Counting Principle

When there are lots of options and you want to determine how many different ways you can get something or do something, the counting principle is the way to go.

Marissa is getting dressed. She has 4 pairs of pants, 5 shirts and 3 different shoes that could all make a decent outfit. How many different outfits can she make that include 1 pair of pants, 1 shirt, and 1 pair of shoes?

The simplest way to get the answer is to use the counting principle. All you have to do is multiply the number of options for each part of the outfit.

Another example:

Mario is out to lunch. He wants a sandwich. There are 3 types of bread, 8 different meats and 6 different cheeses. How many different ways can he build a sandwich?

A different kind of example:

6 people are running together in a 10K race. How many different ways can they finish the race?

6 choices for 1st place
5 choices for 2nd place
4 choices for 3rd place
3 choices for 4th place
2 choices for 5th place
1 choice for 6th place

6 different people can finish first: One of the 6 crosses the finish line.

Now there are 5 people left for second place.

Then 4 people are left for 3rd place.

That leaves 3 people for 4th place.

Next, there are 2 people in line for 5th place.

And there is one person left that finishes last.

The counting principle tells us that we can multiply the number of choices for each place to determine how many different ways the race can end.

We read 6! as 6 factorial.

If we had 5! we would multiply 5 x 4 x 3 x 2 x 1.

If we have 4! we would multiply 4 x 3 x 2 x 1.

The ! is a little shortcut so that you do not have to write out all the numbers less than the given number that you need to multiply.

Let's Review:

The counting principle helps us to determine how many ways we can do something. All you need to do is multiply the number of ways you can do each part to get the number of ways you can do them together.

So if you do the first thing x ways, the second thing y ways and the third thing z ways, then you can do them all together ways.