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Factorial Calculator

Calculate n! (n factorial) for any non-negative integer

Note: Maximum value is 170 due to JavaScript number limitations

Factorial Result:

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Calculation:

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Scientific Notation:

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Related Calculations

Permutations (nPr):

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Combinations (nCr):

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About Factorials

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.

How to Use This Tool

  1. Enter a non-negative integer in the input field (maximum value is 170)
  2. Click the "Calculate Factorial" button to see the result
  3. For permutations and combinations, enter values for n (set size) and r (selection size)
  4. Click either the "Permutations (nPr)" or "Combinations (nCr)" button to calculate

Why We Use This Tool

  • Mathematical calculations: Quickly compute factorials without manual multiplication
  • Educational purposes: Understand the growth rate of factorial sequences
  • Probability calculations: Calculate permutations and combinations for statistical problems
  • Research and analysis: Support work in combinatorics, statistics, and algorithm analysis

Factorial Definition

n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1

By definition, 0! = 1

Examples

5! = 5 × 4 × 3 × 2 × 1 = 120

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

10! = 10 × 9 × 8 × ... × 2 × 1 = 3,628,800

Applications of Factorials

  • Combinatorics: Counting arrangements and selections
  • Probability: Calculating probabilities of events
  • Statistics: Permutations and combinations
  • Calculus: Taylor series expansions
  • Computer Science: Algorithm analysis

Permutations and Combinations

Permutations (nPr): Number of ways to arrange r objects from n distinct objects

nPr = n! / (n - r)!

Combinations (nCr): Number of ways to choose r objects from n distinct objects

nCr = n! / [r! × (n - r)!]

Example: How many ways to arrange 3 books from 5?

5P3 = 5! / (5-3)! = 120 / 2 = 60 arrangements

Example: How many ways to choose 3 books from 5?

5C3 = 5! / [3! × (5-3)!] = 120 / (6 × 2) = 10 selections

Stirling's Approximation

For large values of n, factorials can be approximated using Stirling's formula:

n! ≈ √(2πn) × (n/e)ⁿ

This approximation becomes more accurate as n increases.