A factorial (n!) is the product of all positive integers from 1 to n. For example, 5! = 5×4×3×2×1 = 120.

0! equals 1 by definition. This convention makes many mathematical formulas work consistently.

Why do factorials grow so fast?

Factorials grow exponentially because each step multiplies by an increasingly large number. 20! is already over 2 quintillion.

What are factorials used for?

Factorials are used in permutations, combinations, probability, calculus (Taylor series), and many areas of mathematics and computer science.

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Factorial Calculator

Calculate factorial of a number for combinatorics, probability theory, and mathematical analysis with precision calculation engine. Features large number handling up to thousands of digits, step-by-step calculation display, scientific notation support, and educational explanations of factorial properties. Essential for permutations, combinations, series expansions, and statistical calculations in mathematics, computer science, and probability applications.

How to Use the Factorial Calculator

Use the Factorial Calculator to factorial of a number for combinatorics, probability theory, and mathematical analysis with precision calculation engine. Features large number handling up to thousands of digits, step-by-step calculation display, scientific notation support, and educational explanations of factorial properties. Essential for permutations, combinations, series expansions, and statistical calculations in mathematics, computer science, and probability applications.. Enter your values to get accurate, instant results tailored to your situation.

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Factorial Guide

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Expert Tips

Factorial Formula: n! = n × (n-1) × (n-2) × ... × 2 × 1: 5! = 5 × 4 × 3 × 2 × 1 = 120. 3! = 3 × 2 × 1 = 6. 0! = 1 (by definition). 1! = 1. Factorials grow extremely fast: 10! = 3,628,800, 20! = 2.4 quintillion.
Used in Permutations and Combinations: Permutations (order matters): nPr = n! / (n-r)!. Combinations (order doesn't matter): nCr = n! / (r!(n-r)!). Example: 5 people, choose 3 for committee = 5! / (3! × 2!) = 120 / 12 = 10 ways.
Factorials Grow Faster Than Exponentials: 10! = 3,628,800 vs 2^10 = 1,024. 15! = 1.3 trillion. 20! = 2.4 × 10^18. 100! = 9.3 × 10^157 (more atoms than in universe). Factorial growth astronomical—useful for impossibly large number of arrangements.
Simplify Before Computing Large Factorials: 10! / 8! = (10 × 9 × 8!) / 8! = 10 × 9 = 90. Don't calculate full 10! and 8!. Cancel common terms. 100! / 98! = 100 × 99 = 9,900. Essential for avoiding overflow errors.

Essential Fundamentals — Factorial concepts

Applications

Counting Arrangements: How many ways to arrange n items? n! ways. 3 books: 3! = 6 arrangements (ABC, ACB, BAC, BCA, CAB, CBA). 5 books: 5! = 120 arrangements. Formula accounts for all possible orderings.
Probability Calculations: Deck of cards: 52! possible shuffles = 8 × 10^67 (more than atoms on Earth). Winning lottery: nCr uses factorials to compute odds. Understanding factorials essential for probability and statistics.

Advanced Strategies — Optimization techniques

Computational Strategies

Cancellation Technique: Simplify factorial ratios before computing. 12!/(9!×3!) = (12×11×10)/(3×2×1) = 220. Avoids calculating massive numbers. Essential for combinations and permutations.
Stirling Approximation: For large n: n! ≈ √(2πn) × (n/e)^n. Estimates factorial size without full computation. Used in statistics and probability theory for large sample analysis.

Frequently Asked Questions

What is a factorial?: A factorial (n!) is the product of all positive integers from 1 to n. For example, 5! = 5×4×3×2×1 = 120.
What is 0 factorial?: 0! equals 1 by definition. This convention makes many mathematical formulas work consistently.
Why do factorials grow so fast?: Factorials grow exponentially because each step multiplies by an increasingly large number. 20! is already over 2 quintillion.
What are factorials used for?: Factorials are used in permutations, combinations, probability, calculus (Taylor series), and many areas of mathematics and computer science.