Factorial as Product of Two Factorials/Examples/3
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Example of Factorial as Product of Two Factorials
- $5! = 4 \times 5 \times 6 = \dfrac {6!} {3!}$
Proof
We have that:
| \(\ds 6!\) | \(=\) | \(\ds \paren {3!}!\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3! \paren {3! - 1}!\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3! \, 5!\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 5!\) | \(=\) | \(\ds \dfrac {6!} {3!}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1 \times 2 \times 3 \times 4 \times 5 \times 6} {1 \times 2 \times 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4 \times 5 \times 6\) |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $120$