Factorials at a Glance

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...

bookshelf illustration

We're not just looking for all possible outcomes this time; we want those outcomes sorted in a specific order. We want the books arranged into six different spots.

Let's think about how this works:

We have 6 options for the first book we place on the shelf
Once we've already placed the first book, we 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

Multiplying the options for each slot together gives us the total possible arrangements. This would be 6x5x4x3x2x1=720 .

Luckily, there is a button on our calculator that does the work for us. It looks like an exclamation mark and is called factorial.

The mathematical sign for factorial is "!" but that doesn't mean to shout the number excitedly. "Six factorial" is written 6! and it means 6x5x4x3x2x1=720 .

We can use a factorial to count how many possible ways we could organize our books. In this case, we could organize our books in "six factorial" different arrangements.

Now, let's throw some probability into 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 desired outcome, or one way to arrange these books in alphabetical order. However, there are 6! = 720 ways to arrange them, so the probability that they will be in alphabetical order is:

P = 1/6! = 1/720 ~~0/41%

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

Examples
Exercises

Example 1

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:

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Example 2

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:

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Example 3

You have 9 antique glass vases and want to display 3 on a shelf. How many different ways are there to do this?
Nothin’ like setting them all out and seeing which combo looks best. The first vase is made of an old dark blue glass. If we put that up first, we can either do vase 2, 3, 4, 5, 6, 7, 8, or 9 next.
If we put vase 1 first and vase 2 in the middle, that leaves vase 3, 4, 5, 6, 7, 8, or 9 for the last spot. If we put vase 1 first and vase 3 in the middle, that leaves vase 2, 4, 5, 6, 7, 8, or 9 for the last spot. If we put vase 1 first and vase 4 in the middle, that leaves vase 2, 3, 5, 6, 7, 8, or 9 for the last spot. Writing this out in a list, we get:
It appears that putting vase 1 first leaves 8 choices for the middle vase and 7 choices for the last vase. So that’s 1 × 8 × 7. But that’s just if we put vase 1 first.

We have that same number of options for each vase that we put in first place. Since there are 9 vases, it seems we have 9 × 8 × 7 different ways to arrange 9 vases in 3 spots. Here are the first few possible outcomes: And so on.
Since 9 × 8 = 72 and 72 ×7 = 504, we have 504 choices. Trying them all out? Not a good idea. How about we just buy two more shelves and display them all? Hmmmm, how many different ways could we arrange the 9 vases if we did that? That’s a little light calculating to keep us up tonight.