Factorial Calculator

Calculate factorials (n!) quickly and accurately

Enter a positive integer to calculate its factorial. The factorial of n (written as n!) is the product of all positive integers from 1 to n.

Enter a positive integer between 0 and 170. Factorial is defined for non-negative integers only.

Important: Factorials grow very rapidly. For numbers larger than 170, the result exceeds JavaScript's number precision limits.

Example Calculations

Try these common factorial calculations to understand how the calculator works

Small Number (5!)

small

Calculate factorial of 5 - easy to verify manually

n = 5

Zero Factorial (0!)

zero

Special case: 0! equals 1 by mathematical definition

n = 0

Medium Number (10!)

medium

Calculate factorial of 10 - demonstrates rapid growth

n = 10

Large Number (15!)

large

Calculate factorial of 15 - shows how quickly factorials grow

n = 15

Other Titles
Understanding Factorial Calculator: A Comprehensive Guide
Explore the mathematical concept of factorials, their applications in combinatorics, probability, and various fields of mathematics and science

What is a Factorial? Understanding the Basics

  • Factorials represent the product of consecutive positive integers
  • They are fundamental in combinatorics, probability, and mathematical analysis
  • Factorials have widespread applications in various scientific disciplines
A factorial, denoted by the exclamation mark (!), is a mathematical operation that represents the product of all positive integers from 1 up to a given number n.
For example, 5! (read as 'five factorial') equals 5 × 4 × 3 × 2 × 1 = 120. This simple concept forms the foundation for many advanced mathematical concepts.
The factorial function is defined for non-negative integers, with the special case that 0! = 1 by mathematical convention.
Factorials grow extremely rapidly - much faster than exponential functions. This rapid growth makes them particularly useful in combinatorics for counting permutations and combinations.

Basic Examples

  • 1! = 1 (The factorial of 1 is 1)
  • 2! = 2 × 1 = 2
  • 3! = 3 × 2 × 1 = 6
  • 4! = 4 × 3 × 2 × 1 = 24
  • 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Step-by-Step Guide to Using the Factorial Calculator

  • Learn how to input numbers correctly
  • Understand the calculator's limitations and features
  • Master the interpretation of factorial results
Our factorial calculator is designed to provide instant and accurate calculations for any non-negative integer within computational limits.
Input Guidelines:
  • Number Range: Enter any non-negative integer from 0 to 170. Numbers beyond 170 exceed JavaScript's precision limits.
  • Integer Requirement: The calculator only accepts whole numbers. Decimal inputs will trigger an error message.
  • Special Cases: Remember that 0! = 1 by mathematical definition, and 1! = 1.
Understanding Results:
  • For small numbers (n ≤ 10), results are easy to verify manually and understand intuitively.
  • For larger numbers, results become extremely large very quickly. For example, 20! = 2,432,902,008,176,640,000.
  • The calculator uses BigInt arithmetic to maintain precision for large factorials, ensuring accurate results.

Usage Examples

  • To calculate 8!: Enter 8, click Calculate. Result: 40,320
  • To verify 0!: Enter 0, observe the result is 1 (mathematical convention)
  • To explore large factorials: Try 15! = 1,307,674,368,000
  • To understand growth: Compare 10! = 3,628,800 with 11! = 39,916,800

Real-World Applications of Factorial Calculations

  • Combinatorics and Permutations: Arranging objects and counting possibilities
  • Probability Theory: Computing odds and statistical distributions
  • Computer Science: Algorithm analysis and complexity
  • Scientific Research: Quantum mechanics and statistical models
Factorials serve as powerful tools across numerous practical applications in science, technology, and everyday problem-solving:
Combinatorics and Permutations:
  • Arranging Objects: The number of ways to arrange n distinct objects is n!. This is fundamental in organizing schedules, seating arrangements, and tournament brackets.
  • Combinations Formula: The combination formula C(n,r) = n!/(r!(n-r)!) relies heavily on factorial calculations.
Probability and Statistics:
  • Poker Probabilities: Calculating the odds of different poker hands involves factorial-based combinations.
  • Statistical Distributions: The Poisson distribution and other important statistical models incorporate factorial functions.
Computer Science:
  • Algorithm Complexity: Factorial time complexity O(n!) represents the worst-case scenario for certain algorithms like the traveling salesman problem.
  • Recursive Programming: Factorial calculation is often used as an introductory example for understanding recursion.

Real-World Examples

  • Arranging 8 books on a shelf: 8! = 40,320 different arrangements
  • Choosing a 5-person committee from 10 people: C(10,5) = 10!/(5!×5!) = 252 ways
  • Password with 6 unique characters from 26 letters: 26!/(26-6)! = 165,765,600 possibilities
  • Tournament with 16 teams: Multiple factorial calculations for bracket outcomes

Common Misconceptions and Correct Methods

  • Addressing frequent errors in factorial understanding
  • Clarifying the difference between factorials and other operations
  • Explaining why certain factorial properties exist
Despite their apparent simplicity, factorials are often misunderstood by students. Understanding these common misconceptions helps build a solid foundation:
Misconception 1: Factorial vs. Exponentiation
  • Wrong: Thinking that n! = n^n. This confusion arises because both operations involve repeated multiplication.
  • Correct: n! = n × (n-1) × (n-2) × ... × 1, while n^n = n × n × n × ... × n (n times).
Misconception 2: Zero Factorial
  • Wrong: Believing that 0! = 0 because 'zero times anything equals zero.'
  • Correct: 0! = 1 by mathematical definition. This convention makes many formulas work correctly, particularly in combinatorics.
Misconception 3: Factorial Growth Rate
  • Wrong: Underestimating how quickly factorials grow compared to exponential functions.
  • Correct: Factorials grow much faster than any exponential function. For large n, n! >> a^n for any constant a.

Correction Examples

  • Common Error: 3! = 3^3 = 27. Correct: 3! = 3 × 2 × 1 = 6
  • 0! = 1, not 0. This allows C(n,0) = n!/0! = 1 to work correctly
  • Growth comparison: 10! = 3,628,800 while 2^10 = 1,024

Mathematical Derivation and Examples

  • Historical development of factorial notation and concepts
  • Mathematical properties and relationships
  • Advanced applications in higher mathematics
The mathematical foundation of factorials extends beyond simple multiplication, connecting to advanced concepts in analysis and number theory:
Mathematical Properties:
  • Recursive Definition: n! = n × (n-1)! for n > 0, with 0! = 1 as the base case.
  • Stirling's Approximation: For large n, n! ≈ √(2πn) × (n/e)^n, providing an excellent approximation.
  • Ratio Property: (n+1)!/n! = n+1, which simplifies many combinatorial calculations.
Advanced Applications:
  • Taylor Series: Many functions can be expressed as infinite series involving factorials in their denominators.
  • Gamma Function: The gamma function Γ(n) = (n-1)! extends factorials to complex numbers and non-integers.
  • Probability Distributions: The Poisson distribution uses e^(-λ) × λ^k / k! to calculate probabilities.

Advanced Examples

  • Stirling's approximation for 10!: √(20π) × (10/e)^10 ≈ 3,598,696 vs actual 3,628,800
  • Taylor series for e^x: 1 + x + x²/2! + x³/3! + x⁴/4! + ...
  • Poisson probability: P(X=3) = e^(-2) × 2³/3! for λ=2
  • Gamma function: Γ(5) = 4! = 24, Γ(1/2) = √π