Factorial as Product of Three Factorials

Theorem

This general pattern can be used to find a factorial which is the product of three factorials:

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

while there are instances of factorials which do not fit that pattern.

Proof

\(\ds \paren {\paren {n!}!}!\)

\(=\)

\(\ds \paren {n!}! \times \paren {\paren {n!}! - 1}!\)

Factorial as Product of Two Factorials

\(\ds \)

\(=\)

\(\ds n! \times \paren {n! - 1}! \times \paren {\paren {n!}! - 1}!\)

Factorial as Product of Two Factorials

$\blacksquare$

Examples

$10!$ as Product of $3$ Factorials

$10! = 7! \times 5! \times 3!$

$16!$ as Product of $3$ Factorials

$16! = 14! \times 5! \times 2!$