What is a factorial?

You may have encountered factorials when studying probability. But what exactly is a factorial? The Fastmath team is here to explain.

A factorial represents the product of all positive integers from $1$ to $n$. It is written by placing an exclamation mark, $!$, after a number or variable, such as $5!$ or $n!$, read as “five factorial” or “n factorial.”

Although a factorial is the product of the positive integers from $1$ to $n$, there is another important value to remember: $\bold{0! = 1}$. This is particularly useful in mental arithmetic. In Fastmath, factorials are available at Level 3. Under the rules of the 2023 Student Arts and Crafts Competition, they are permitted at secondary-school level.

Examples of factorial calculations

Useful factorial identities

These identities appear frequently in mental arithmetic games. Memorizing them can give you more ways to reach an answer. Here are ten useful examples collected by the Fastmath team:

We also have two more techniques to share:

Factorials do more than shorten long strings of multiplication. They can also create numbers that might initially seem impossible to obtain. We hope you can put these techniques to use in your next calculation or mental arithmetic game.