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Understanding Factorials

The factorial of a non-negative integer 'n', denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, the factorial of 0 (0!) is defined as 1. Factorials grow very rapidly as n increases.

The concept and notation for factorials evolved over time. While the product of consecutive integers appeared in earlier mathematical works (e.g., in Indian mathematics around the 12th century related to permutations), the notation "n!" was introduced by the French mathematician Christian Kramp in 1808. Factorials became fundamental in combinatorics, probability theory, and calculus (e.g., in Taylor series).

Factorials are primarily used in mathematics, particularly in combinatorics to count permutations (the number of ways to arrange n distinct objects) and combinations. They also appear in probability calculations, calculus (especially in series expansions like Taylor series for functions like e^x), and various areas of physics and computer science algorithms.