# Factorial as Product of Two Factorials

## Theorem

Apart from the general pattern, following directly from the definition of the factorial:

$\paren {n!}! = n! \paren {n! - 1}!$

the only known factorial which is the product of two factorials is:

$10! = 6! \, 7!$

## Proof

 $\displaystyle 10!$ $=$ $\displaystyle 7! \times 8 \times 9 \times 10$ Definition of Factorial $\displaystyle$ $=$ $\displaystyle 7! \times 2 \times 4 \times 3 \times 3 \times 2 \times 5$ $\displaystyle$ $=$ $\displaystyle 7! \times 2 \times 3 \times 4 \times 5 \times 6$ $\displaystyle$ $=$ $\displaystyle 6! \, 7!$ Definition of Factorial

$\blacksquare$

## Examples

### Factorial of Factorial of $3$

$5! = 4 \times 5 \times 6 = \dfrac {6!} {3!}$