∑

Factorial Notation (n!)

Elementary

Definition

n! (n factorial) is the product of all positive integers from 1 to n. Essential in permutations and combinations.

n! = n × (n-1) × (n-2) × ... × 2 × 1

Definition of factorial

0! = 1

Zero factorial (by definition)

n! = n × (n-1)!

Recursive definition

n! ≈ √2π n ((n)/(e))ⁿ

Stirling's approximation

Example 1

Calculate 5!

Example 2

Calculate 7!/5!

Discovered by: Christian Kramp (1808)

First used the ! symbol, which became standard notation.

Permutations, combinations

Binomial, Poisson distributions

Algorithm complexity