Factorial Notation

The Meaning of $n!$

The product of the first $n$ natural numbers is given a compact symbol: $n!$, read as "n factorial".

So $1! = 1$, $2! = 1 \times 2 = 2$, $3! = 1 \times 2 \times 3 = 6$, $4! = 1 \times 2 \times 3 \times 4 = 24$, and so on. The factorial grows very quickly — $5!$ is already $120$, and $7!$ is $5040$.

The Special Case: $0!$

We define $0! = 1$. This is not a product of natural numbers; it is a convention that makes many formulas in permutations and combinations work smoothly. Without it, formulas like $P(n, n) = n!$ would break when $n = 0$.

Recursive Property of Factorials

For any natural number $n$, we can write $n!$ in terms of $(n-1)!$:

This is the most useful property of factorials. It lets us "peel off" factors one at a time.

For example:

More generally, for $n \ge 2$:

For $n \ge 3$:

And so on, until we reach $1!$. …