**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.

## Factorials Practice:

Maisy is working the counter at Shmaskin Robbins. A hungry customer orders a triple scoop ice cream cone with strawberry, chocolate, and vanilla ice cream. How many different ways could she stack the ice cream flavors on top of each other? | |

You could answer the question by listing all of the possible orders of the ice cream (top to bottom). - Chocolate-Strawberry-Vanilla
- Chocolate-Vanilla-Strawberry
- Strawberry-Vanilla-Chocolate
- Strawberry-Chocolate-Vanilla
- Vanilla-Strawberry-Chocolate
- Vanilla-Chocolate-Strawberry
Or, use factorials to get the answer much faster: 3! = 3 × 2 × 1 = 6 | |

Now, what's the probability that chocolate will be on top? | |

Two of these combinations have chocolate on top - Chocolate-Vanilla-Strawberry
- Chocolate-Strawberry-Vanilla
So the probability is 2 out of the 6 possible combinations: | |

Tom, Tristan, Luca, and Pablo are lined up and ready to be picked for kickball teams. How many different ways can they be picked from first to last? | |

We could list the ways, but that isn't necessary now that we know about awesome factorials. The number of ways to pick the four boys is 4! = 4 × 3 × 2 × 1 = 24. | |

If all boys were picked in a random order, what is the probability that Tom will luck out and be picked first? | |

To answer the probability question we need to look at all the ways that Tom could be picked first. Here they are: - Tom-Tristan-Luca-Pablo
- Tom-Tristan-Pablo-Luca
- Tom-Pablo-Luca-Tristan
- Tom-Pablo-Tristan-Luca
- Tom-Luca-Pablo-Tristan
- Tom-Luca-Tristan-Pablo
If we fix Tom as the kid picked first, and arrange the other three, we find 6 different ways, which is 3!. 3! = 3 × 2 × 1 = 6 So the probability that Tom is picked first is 6 out of a possible 24 combinations. The probability is: | |

How many ways can you arrange 10 different items?

Your mom has framed photos of you from 1st grade to 7th. She is going to hang them in the hall in a long row. How many ways can she display them?

If she hangs these randomly, what is the probability that they will be in chronological order from left to right or right to left.

Hint

there are two ways to do this

Answer