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:

$10! = 6! \, 7!$

$\ds 10!$

$=$

$\ds 7! \times 8 \times 9 \times 10$

Definition of Factorial

$\ds $

$=$

$\ds 7! \times \paren {2 \times 4} \times \paren {3 \times 3} \times \paren {2 \times 5}$

$\ds $

$=$

$\ds 7! \times 2 \times 4 \times 3 \times \paren {3 \times 2} \times 5$

$\ds $

$=$

$\ds 7! \times 2 \times 3 \times 4 \times 5 \times 6$

$\ds $

$=$

$\ds 6! \, 7!$

Definition of Factorial

$\blacksquare$

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