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
\(\ds 10!\) | \(=\) | \(\ds 7! \times 8 \times 9 \times 10\) | 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 |
$\blacksquare$
Examples
Factorial of Factorial of $3$
- $5! = 4 \times 5 \times 6 = \dfrac {6!} {3!}$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $10$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10$