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