What is factorial?

Factorial multiplies a positive whole number by every positive whole number below it, down to 1. Written with an exclamation mark: n!

Examples:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 4! = 4 × 3 × 2 × 1 = 24
• 3! = 3 × 2 × 1 = 6
• 1! = 1
• 0! = 1 (by definition a convention that makes formulas work cleanly)

Where factorials appear: whenever you count arrangements.

How many ways can 5 people stand in a line? First position: any of 5. Second: any of remaining 4. Third: any of 3. And so on: 5 × 4 × 3 × 2 × 1 = 120 = 5!

Factorials grow explosively fast:

• 10! = 3,628,800
• 20! = 2,432,902,008,176,640,000

This rapid growth is why problems involving arrangements become computationally enormous a key insight in algorithms and cryptography.