# Understanding the Factorial (!) in Mathematics and Statistics

In mathematics, symbols that have certain meanings in the English language can mean very specialized and different things. For example, consider the following expression:

3!

No, we did not use the exclamation point to show that we’re excited about three, and we shouldn’t read the last sentence with emphasis. In mathematics, the expression 3! is read as "three factorial" and is really a shorthand way to denote the multiplication of several consecutive whole numbers.

Since there are many places throughout mathematics and statistics where we need to multiply numbers together, the factorial is quite useful. Some of the main places where it shows up are combinatorics and probability calculus.

## Definition

The definition of the factorial is that for any positive whole number n, the factorial:

n! = n x (n -1) x (n - 2) x . . . x 2 x 1

## Examples for Small Values

First we will look at a few examples of the factorial with small values of n:

• 1! = 1
• 2! = 2 x 1 = 2
• 3! = 3 x 2 x 1 = 6
• 4! = 4 x 3 x 2 x 1 = 24
• 5! = 5 x 4 x 3 x 2 x 1 = 120
• 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
• 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040
• 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320
• 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362880
• 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3628800

As we can see the factorial gets very large very quickly. Something that may seem small, such as 20! actually has 19 digits.

Factorials are easy to compute, but they can be somewhat tedious to calculate. Fortunately, many calculators have a factorial key (look for the ! symbol). This function of the calculator will automate the multiplications.

## A Special Case

One other value of the factorial and one for which the standard definition above does not hold is that of zero factorial. If we follow the formula, then we would not arrive at any value for 0!. There are no positive whole numbers less than 0. For several reasons, it is appropriate to define 0! = 1. The factorial for this value shows up particularly in the formulas for combinations and permutations.