Calculate the factorial of any positive integer. Understand factorials, formulas, permutations, and mathematical combinations.
The factorial function, denoted n!, is a fundamental mathematical operation defined as the product of all positive integers from 1 to n. Factorial(5) = 5 × 4 × 3 × 2 × 1 = 120. Factorial(0) = 1 by mathematical convention. Factorials grow explosively fast — factorial(20) already exceeds 2 quintillion, and factorial(100) has 158 digits.
Factorials appear throughout mathematics, statistics, and computer science. In combinatorics, n! counts the number of distinct ways to arrange n objects in a sequence (permutations). The binomial coefficient C(n,k) = n! / (k! × (n-k)!) uses factorials to count combinations. Taylor series expansions of exponential and trigonometric functions use factorials as denominators.
Probability calculations frequently require factorials. The number of ways to arrange k items chosen from n distinct items without repetition is P(n,k) = n!/(n-k)!. The multinomial coefficient — counting arrangements when items repeat — extends the binomial formula using factorials of each repeat count. The Poisson distribution, normal distribution series expansions, and Bayesian inference calculations all involve factorial operations.
Statistical significance tests like Fisher's exact test compute exact p-values using factorials to enumerate all possible configurations of a contingency table. Understanding factorial growth rates is critical for assessing computational complexity — algorithms with O(n!) time complexity (like naive traveling salesman) become computationally infeasible extremely quickly as n increases.
Standard 64-bit floating-point numbers can only represent factorials exactly up to 20! (which equals 2,432,902,008,176,640,000). Beyond this, IEEE 754 double-precision arithmetic loses exact integer precision. Computing 21! in standard floating-point gives an approximation rather than the exact value.
This calculator uses arbitrary-precision integer arithmetic to compute exact factorial values for any input. For display purposes, results beyond a certain digit count are shown in scientific notation with the exact digit count stated. The Stirling approximation ln(n!) ≈ n×ln(n) - n + 0.5×ln(2πn) provides excellent estimates for large n without exact computation.
Factorial n! is the product of all positive integers from 1 to n. Zero factorial equals 1 by definition (the empty product convention) and because it is required for consistency in combinatorial formulas like C(n,0)=1 — there is exactly one way to choose zero items from any set.
This calculator uses arbitrary-precision integer arithmetic and can compute factorials of very large numbers. However, display may truncate extremely large results to scientific notation showing the magnitude. Practically, factorials up to 10,000! are computable though the result has 35,659 digits.
Factorials grow faster than exponential functions. By Stirling's approximation, ln(n!) ≈ n·ln(n) - n, meaning n! grows roughly as (n/e)^n × √(2πn). Factorial(20) ≈ 2.4 × 10^18, factorial(100) ≈ 9.3 × 10^157, factorial(1000) has 2568 digits.
The Gamma function Γ(n) extends factorial to non-integer values: Γ(n) = (n-1)! for positive integers. For non-integers, Γ(0.5) = √π. This extension allows factorial-like computations for real and complex numbers, enabling factorials to appear in continuous probability distributions like the chi-squared distribution.
Yes extensively. Algorithms that enumerate all permutations have O(n!) complexity. Hash functions, random number generators, and error-correcting codes use modular factorial arithmetic. The n-queens problem, Hamiltonian path problem, and certain dynamic programming recurrences involve factorial relationships.