Expert Guide to Factorials in 2026
The factorial is a cornerstone concept in mathematics, appearing everywhere from high school algebra to quantum mechanics. Denoted by n!, it represents the product of all positive integers up to n. While the definition is simple, the applications are vast.
Factorial Examples (n!)
The table below shows factorial values for common integers. Factorials grow extremely rapidly—notice how 10! is already over 3.6 million, and 20! exceeds 2.4 quintillion. This exponential growth makes factorials crucial in combinatorics, probability, and algorithm analysis.
| n | n! (Factorial) | Scientific Notation | Digits | Common Use |
|---|---|---|---|---|
| 0 | 1 | 1.00 × 10⁰ | 1 | Base case, empty arrangements |
| 1 | 1 | 1.00 × 10⁰ | 1 | Single item arrangement |
| 2 | 2 | 2.00 × 10⁰ | 1 | Two items: 2 ways |
| 3 | 6 | 6.00 × 10⁰ | 1 | Three items: 6 ways |
| 4 | 24 | 2.40 × 10¹ | 2 | Four items: 24 ways |
| 5 | 120 | 1.20 × 10² | 3 | Common in probability |
| 6 | 720 | 7.20 × 10² | 3 | Dice combinations |
| 7 | 5,040 | 5.04 × 10³ | 4 | Week permutations |
| 8 | 40,320 | 4.03 × 10⁴ | 5 | Chess piece arrangements |
| 10 | 3,628,800 | 3.63 × 10⁶ | 7 | 10 items: 3.6M ways |
| 12 | 479,001,600 | 4.79 × 10⁸ | 9 | Calendar months |
| 15 | 1,307,674,368,000 | 1.31 × 10¹² | 13 | Large permutations |
| 20 | 2,432,902,008,176,640,000 | 2.43 × 10¹⁸ | 19 | Combinatorial explosion |
Advanced Applications
**Cryptography & Security**: The massive size of factorials makes brute-force attacks on permutations computationally infeasible.
**Artificial Intelligence**: Factorials appear in probability distributions used in Bayesian networks and machine learning algorithms.
**Statistical Mechanics**: In physics, calculating the number of microstates in a system (entropy) often involves factorials of huge numbers.
Related Math Calculators
Factorials are closely related to other mathematical concepts:
- Permutations & Combinations Calculator: Calculate the number of ways to arrange or select items. Permutations use factorials directly: P(n,r) = n! / (n-r)!
- GCD & LCM Calculator: Find the greatest common divisor and least common multiple of numbers, useful for simplifying fractions and solving number theory problems.
- Powers & Roots Calculator: Calculate exponents and roots, which often appear alongside factorials in series expansions and probability calculations.
Frequently Asked Questions
Q:What is a factorial?
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. It represents the number of ways to arrange n distinct objects into a sequence.
Q:Why is 0 factorial equal to 1?
0! = 1 is a fundamental convention in mathematics. It ensures consistency in combinatorial formulas (like nCr), allows recursive definitions (n! = n × (n-1)!) to work for n=1, and aligns with the Gamma function where Γ(1) = 1. Conceptually, there is exactly one way to arrange zero objects: by doing nothing.
Q:What is the Gamma Function?
The Gamma function, denoted by Γ(z), extends the concept of factorial to complex numbers. For any positive integer n, Γ(n) = (n-1)!. This allows for the calculation of 'factorials' for non-integers and negative numbers (except negative integers), which is crucial in advanced physics and engineering.
Q:What is the largest factorial this calculator can handle?
Unlike standard calculators limited to 170!, this tool uses BigInt technology to compute factorials for n up to 5000 and beyond. It provides exact integer results without rounding errors, essential for high-precision cryptography and scientific computing contexts in 2026.
Q:What are the real-world applications of factorials?
Factorials are used in permutations and combinations (calculating odds in lottery or poker), probability theory, Taylor series expansions in calculus (e.g., for calculating sine and cosine), number theory, and analyzing the time complexity of algorithms (O(n!)) in computer science. To calculate permutations and combinations, use our Permutations & Combinations Calculator (/math/permutations-combinations-calculator).