# Factorial Notation

$n!$ is the product of the first $n$ positive integers for $n\ge 1$.
$n!=n(n-1)(n-2)(n-3) \cdots \times 3 \times 2 \times 1$
$n!$ is read “$n$ factorial”.

For example, the product $5 \times 4 \times 3 \times 2 \times 1$ can be written as $5!$.
Notice that $5 \times 4 \times 3$ can be written using factorial numbers only as
$5 \times 4 \times 3 = \dfrac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = \dfrac{5!}{2!}$

You can also notice that:
\begin{align} n! &= n(n-1)(n-2)(n-3) \cdots \times 3 \times 2 \times 1 \\ &= n(n-1)! \\ &= n(n-1)(n-2)! \\ &= n(n-1)(n-2)(n-3)! \\ &= \cdots \\ \end{align}

Using this rule of factorial:
\begin{align} 1! &= 1 \times 0! \\ 0! &= 1 \\ \end{align}

## Example 1

Evaluate $5!$.

\begin{align} \displaystyle 5! &= 5 \times 4 \times 3 \times 2 \times 1 \\ &= 120 \end{align}

## Example 2

Evaluate $\dfrac{6!}{4!}$.

\begin{align} \displaystyle \dfrac{6!}{4!} &= \dfrac{6 \times 5 \times 4!}{4!} \\ &= 6 \times 5 \\ &= 30 \end{align}

## Example 3

Evaluate $\dfrac{7!}{4! \times 3!}$.

\begin{align} \displaystyle \dfrac{7!}{4! \times 3!} &= \dfrac{7 \times 6 \times 5 \times 4!}{4! \times 3!} \\ &= \dfrac{7 \times 6 \times 5}{3!} \\ &= \dfrac{7 \times 6 \times 5}{3 \times 2 \times 1} \\ &= 35 \end{align}