What is a Factorial?

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n.

Special cases:

• 0! = 1 (by mathematical definition)
• 1! = 1

Examples

• 2! = 2 × 1 = 2
• 3! = 3 × 2 × 1 = 6
• 4! = 4 × 3 × 2 × 1 = 24
• 5! = 5 × 4 × 3 × 2 × 1 = 120

Stirling's Approximation

For large values of n, computing the exact factorial becomes difficult due to the rapid growth. Stirling's formula provides an approximation:

where e is the mathematical constant approximately equal to 2.71828.

Applications of Factorials

• Combinations and Permutations: Used to calculate the number of ways to arrange or select items.
• Probability Theory: Essential for calculating various probability distributions.
• Series Expansions: Used in Taylor series and other mathematical expansions.
• Number Theory: Important in various number theory problems and formulas.
• Statistical Distributions: Used in defining distributions like the Poisson distribution.

Recursive Definition

Factorials can be defined recursively as:

This recursive definition is often used in programming implementations.