The factorial of n, written n!, is the product 1 × 2 × 3 × ... × n. For example, 5! = 120. By convention, 0! = 1. Factorials grow quickly: 10! is already 3,628,800.

Factorials count arrangements. The number of ways to order n distinct objects is n!. That makes them central to permutations and combinations. The binomial coefficient, which counts combinations, is written using factorials: n! / (r!(n − r)!).

Factorials appear in probability, series expansions, and the gamma function, which extends the concept to non-integers. Stirling's approximation estimates large factorials. In computer science, factorial growth is a benchmark for algorithmic complexity. The notation is compact, but the numbers explode fast.

• Product of all positive integers up to n
• 0! is defined as 1
• Counts arrangements (permutations)
• Used in combinations, probability, and series