Study Guide

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...

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.