# Factorial Notation

Factorial Notations

$n!$ 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 $5!$.

### Example 2

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

### Example 3

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