Factorial Calculator

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.

0! equals 1. This is a mathematical convention that makes many formulas work correctly, including combinations and permutations. It represents the number of ways to arrange zero objects, which is one way (do nothing).

Factorials grow faster than exponential functions. Each successive factorial multiplies by an increasing number: 10! ≈ 3.6 million, 20! ≈ 2.4 quintillion. This rapid growth limits practical calculations to relatively small numbers.

Factorials are essential in combinatorics for calculating permutations and combinations, probability theory, Taylor series in calculus, and various areas of statistics and computer science algorithms.

Permutations are calculated using factorials. The number of ways to arrange n objects is n!. To find permutations of r objects from n objects, use P(n,r) = n!/(n-r)!. For example, arranging 3 letters from 5 is 5!/2! = 60 ways.

Permutations consider order (ABC vs BAC are different), calculated as n!/(n-r)!. Combinations ignore order (ABC = BAC), calculated as n!/(r!(n-r)!). For example, selecting 2 from 4 letters gives 12 permutations but only 6 combinations.

Factorials are not defined for negative integers in standard mathematics. The factorial function only applies to non-negative integers (0, 1, 2, 3...). For negative or non-integer values, the gamma function is used as an extension.

Standard calculators can compute factorials up to about 170! before encountering overflow errors. 170! ≈ 7.26 × 10^306. Beyond this, specialized arbitrary-precision libraries are needed. For most practical applications, factorials above 20! are rarely needed.

Factorials are fundamental in probability for counting outcomes. The probability of events often involves combinations C(n,r) = n!/(r!(n-r)!). For example, the probability of a specific poker hand uses factorials to count total possible hands and favorable outcomes.

100! is approximately 9.33 × 10^157, a number with 158 digits. The exact value is 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000.