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What Is a Factorial?

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  • 2:10 Fundamental Counting Principle
  • 2:41 Factorial
  • 3:28 Example #1
  • 4:11 Example #2
  • 4:50 Lesson Summary
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Lesson Transcript
Instructor: Luke Winspur

Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

Maybe it's because I'm a math teacher, but when I watched the Olympics I found myself thinking about how many different ways the swimmers could have finished the race. In this video, you'll learn the answer to this question, why it's important and how it lead to the invention of the mathematical operation called the factorial.

The Olympic Swim Problem

As I'm writing this, the Summer Olympics in London are happening, and I've been watching a lot of the swimming events. I'm not much of a swimmer myself, so it amazes me how easily the athletes can glide through the water. It's like they're dolphins or something. Anyway, maybe it's because I'm a math teacher, but I often find myself thinking: how many different ways could all eight of these people finish this race? Maybe they'll end up finishing in this order, or maybe like this or maybe even this. We just thought up three different ways they could finish, but there are probably tons more. How many do you think?

If we tried to actually come up with every single outcome, one at a time, it would take days, and we'd probably kill a tree with all the paper we used up. So, let's see if we can come up with a shortcut.

Let's say that the race is still happening, and no one has finished yet. That means that there are eight different possible winners. Maybe Alex will win, or maybe Chris - we don't know; it could be anyone. But once the winner finishes (whoever it is), now there are only seven different people that might get second.

Continuing this pattern would mean that, at this point, there are only six different people that could get third. After third place finishes, there are only five different people that could get fourth. That means four different people could get fifth, three different people could get sixth and there are two different people that might get seventh. Once the first seven people have finished, there's only one person left that might get last, which means that to answer the question of how many different ways this race could end up, we simply have to multiply 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 to find out that there are 40,320 different possible ways for this race to end.

The fundamental counting principle: multiply by how many possibilities exist each step of the way
Fundamental counting principle

The Fundamental Counting Principle

What we just did is actually called the fundamental counting principle. It tells us that when we are presented with a long list of possibilities, and we're trying to count up how many total outcomes might happen, we can simply multiply by how many possibilities there are at each step of the way. It won't always be the case that the numbers we're multiplying by are decreasing by 1 each step, but it does end up happening a lot. It happens so often, in fact, that mathematicians decided to give the operation its own name, factorial.

Factorial

We denote factorial with an exclamation point, and it simply tells us to multiply any natural number by all the natural numbers that are smaller than it. If we're asked to evaluate 5!, I simply have to do 5 * 4 * 3 * 2 * 1, and I get 120. 9! is 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 =362,880. 20! is 20 * 19 * 18 * 17 * 16 all the way to 3 * 2 * 1 and I get this outrageous number 24329... on and on and on. As you can see, factorials get real big real quick.

Along with helping us answer questions like 'how many different ways could these people finish the race?', a factorial can also be used for questions having to do with arranging things.

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