# Factorials

**Factorials** help count things like *arrangements* of items or *order* of events.

Let's say that you have six books to organize on your bookshelf...

Use a factorial to count how many possible ways you could organize your books. In this case, you could organize your books in "six factorial" different arrangements.

The mathematical sign for factorial is "!" (that doesn't mean to shout the number excitedly)

"Six factorial" = 6! =

Think about how this works:

- You have 6 options for the first book you place on the shelf
- Once you've already placed the first book, you have 5 remaining options for the second book
- Then, 4 options for the third
- Then, 3 options for the fourth
- Then, 2 options for the fifth
- Then, only 1 option for the sixth

When you have a factorial, multiply all positive integers less than or equal to the given number. Another example:

Now, back to our book example. If these books are randomly arranged, what is the probability that they will be in alphabetical order?

Answer: there is only one way to arrange these in alphabetical order. However, there are 6! = 720 ways to arrange them, so:

**Look Out**: factorials give you the number of ways to arrange ALL of the items in a group, not just a portion of them.