Factorial as Product of Two Factorials

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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 $3$

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


Sources