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

What is the maximum range for factorial calculation?

JavaScript's double-precision floating-point can represent up to approximately 1.7×10^308, so 170!≈7.26×10^306 is the largest factorial that can be precisely represented; 171! overflows to Infinity.

What are the applications of factorial?

Factorials are widely used in permutation and combination calculations, such as P(n,k)=n!/(n-k)! and C(n,k)=n!/(k!(n-k)!). Also used in probability theory, Taylor series expansion, and other mathematical fields.

🔢 Factorial Calculator

Calculate n! for 0 ≤ n ≤ 170

Positive integer n (0-170)

10!

3,628,800

10! = 10×9×8×7×6×5×4×3×...

Common Factorial Values

n	n!
0!	1
1!	1
2!	2
3!	6
4!	24
5!	120
6!	720
7!	5,040
8!	40,320
9!	362,880
10!	3,628,800
12!	479,001,600
15!	1,307,674,368,000
20!	2,432,902,008,176,640,000

Factorial Calculator Guide

This calculator computes n factorial (n!). Enter a non-negative integer n to calculate n!, supporting the range 0 to 170.

Factorial is an important mathematical concept widely used in permutations, combinations, probability theory, and Taylor series. n! = n × (n-1) × (n-2) × ... × 1, with 0! = 1 by definition.

Properties and Applications of Factorials

Basic properties:

- 0! = 1 (by definition) - 1! = 1 - n! = n × (n-1)! - Factorials grow extremely fast: 10! = 3,628,800, 20! ≈ 2.43×10^18

Applications: 1. Permutations: P(n,n) = n! 2. Combinations: C(n,r) = n! ÷ (r! × (n-r)!) 3. Probability calculations 4. Taylor series expansion 5. Stirling's approximation: n! ≈ √(2πn) × (n/e)^n

Note: 170! ≈ 7.26 × 10^306, which exceeds double-precision floating point limits.

This calculator is for reference only. Please verify results independently.

FAQ

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Factorial Calculator Guide

The Factorial Calculator quickly computes the factorial of a positive integer n, denoted n!. A factorial is the product of all positive integers less than or equal to n. This tool supports the range 0 to 170 (limited by JavaScript floating-point precision) and automatically displays large results in scientific notation.

Factorial is a fundamental concept in combinatorics, with important applications in permutations and combinations, probability and statistics, Taylor series expansion, and more. Understanding factorial concepts and properties is a necessary foundation for studying advanced mathematics. This tool also provides a quick reference table of common factorial values.

Definition and Properties of Factorial

【Definition】n! = n × (n-1) × (n-2) × ... × 2 × 1
【Special Values】0! = 1 (by convention), 1! = 1
【Recurrence】n! = n × (n-1)!
【Growth Rate】Factorial grows faster than exponential functions (Stirling's approximation)
【Stirling's Formula】n! ≈ √(2πn) × (n/e)^n
【Range Limit】JavaScript double precision can accurately represent up to 170!

Practical Examples

📋

【Example 1】Calculate 5! 5! = 5 × 4 × 3 × 2 × 1 = 120 Application: Number of ways to arrange 5 distinct elements 【Example 2】Calculate 10! 10! = 3,628,800 Application: Number of ways to line up 10 people 【Example 3】Calculate 20! 20! = 2,432,902,008,176,640,000 Application: Part of the total possible shuffles of a deck of 52 cards 【Example 4】Calculate 100! 100! ≈ 9.33 × 10^157 This is a 158-digit enormous number!

Important Notes

💡

- Factorial is only defined for non-negative integers; decimals and negative numbers are not supported (the Gamma function extends to real numbers) - 170!≈7.26×10^306 is the largest factorial JavaScript can precisely represent - 0! = 1 is a mathematical convention (empty product), not a calculation error - n! grows extremely fast: 70! already exceeds 10^100 (a googol) - Large factorials have important applications in cryptography (e.g., RSA algorithm security is based on the difficulty of large number factorization)

Application Scenarios

- Permutations and combinations: P(n,r)=n!/(n-r)! for permutation count
- Probability theory: Core component of binomial distribution, Poisson distribution formulas
- Taylor series: Power series expansions of e^x, sin(x), cos(x) include factorial terms
- Statistics: Multinomial coefficients, hypergeometric distribution calculations
- Cryptography: Large number operations in RSA encryption key generation

Factorial and the Gamma Function

Factorial can be extended to non-integer values through the Gamma function Γ(n). For positive integers n, the relationship is: Γ(n) = (n-1)!. The Gamma function is also defined on the complex plane and is an important special function in mathematical analysis.

Some useful fractional factorial values (via the Gamma function): (1/2)! = Γ(3/2) = √π/2 ≈ 0.886 This result appears in the probability density function of the normal distribution, reflecting the connection between factorial concepts and continuous mathematics.

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