Factorial Calculator

Calculate the factorial (n!) of any whole number. Essential for probability, combinatorics, permutations, and combinations calculations.

Calculator

Result

Input (n)
-
Factorial (n!)
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Formula & Guide

Formula

!

Factorial Definition

n! = n × (n-1) × ... × 2 × 1

Product of all positive integers from 1 to n

0

Special Case

0! = 1

Zero factorial is defined as 1

Formula Variables

n

Number

A non-negative integer (0, 1, 2, 3, ...)

n!

Factorial

The product of all positive integers from 1 to n, or 1 if n = 0

Example Scenario

Number (n)

5

1

Understand the Operation

  • 5! means multiplying all positive integers from 1 to 5

Factorial is the product of descending positive integers

2

Perform the Calculation

  • 5! = 5 × 4 × 3 × 2 × 1
  • = 20 × 3 × 2 × 1
  • = 60 × 2 × 1
  • = 120 × 1
= 120

Additional Examples

Small Factorial

Number: 4

Calculation

4! = 4 × 3 × 2 × 1

Result

4! = 24

Zero Factorial

Number: 0

Definition

0! = 1

Result

0! = 1

Characteristics of Factorials

Rapid Growth

Factorials grow extremely fast. 10! = 3,628,800, and 20! has 19 digits. This rapid growth makes them useful but computationally expensive.

Combinatorics

Factorials are fundamental in counting problems, permutations, and combinations. They represent the number of ways to arrange n distinct objects.

Probability Applications

Used extensively in probability theory to calculate the number of possible outcomes and arrangements in random experiments.

Mathematical Properties

Factorials have important properties: n! = n × (n-1)!, and they're related to the gamma function for non-integer values.

Important Notes

  • 0! is defined as 1, which is a fundamental convention in mathematics.
  • Factorials are only defined for non-negative integers. Negative numbers and decimals don't have standard factorials.
  • Factorials grow extremely fast, so calculations for large numbers (n > 170) may exceed standard number representations.
  • Factorials are used in permutations (nPr = n!/(n-r)!) and combinations (nCr = n!/(r!(n-r)!)).
  • The gamma function extends factorials to real and complex numbers: Γ(n+1) = n! for positive integers.

Frequently Asked Questions

Find answers to common questions about factorial calculations.