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Factorial Calculator

Calculate the factorial of any non-negative integer with step-by-step working.

n! calculatorCounting and probabilityUpdated May 2026

Use a non-negative whole number from 0 to 170.

Quick values

BigInt calculation · Step view · Updated May 2026

Factorial result

10! = 3,628,800

Product of all positive whole numbers from n down to 1.

Input value

10

The n in n!.

Number of digits

7

Length of the exact result.

Formula used

n! = n × (n − 1) × ... × 1

Exact value

3628800

Full result shown.

Expanded form

10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800

10! means multiplying 10 by every positive whole number below it.
10! has 7 digit(s).
Factorials are useful for counting arrangements and probability outcomes.
Factorial values grow very quickly as n increases.
Use combinations when order does not matter and permutations when order matters.
This calculator uses BigInt for exact integer factorial results.

Factorial Formula

Factorial

n! = n × (n − 1) × ... × 1

Zero factorial

0! = 1

Recursive form

n! = n × (n − 1)!

Permutation formula

nPr = n! ÷ (n − r)!

Combination formula

nCr = n! ÷ (r! × (n − r)!)

Example

5! = 5 × 4 × 3 × 2 × 1

Variable Explanations

n

The non-negative whole number used in n!.

!

The factorial symbol.

n!

The product of all positive integers from n down to 1.

0!

Defined as 1.

nPr

Permutations: ordered arrangements.

nCr

Combinations: unordered selections.

How Factorials Work

Factorials multiply descending positive integers.
Each larger n adds another multiplication step.
Factorials grow faster than powers for many common values.
0! equals 1, which keeps counting formulas consistent.
Factorials are central to permutations and combinations.
Large factorials can have hundreds of digits.

Worked Examples

Find 5!

Formula: n! = n × (n − 1) × ... × 1

Substitution: 5 × 4 × 3 × 2 × 1

Answer: 120

Multiply all positive whole numbers from 5 down to 1.

Find 0!

Formula: 0! = 1

Substitution: Defined value

Answer: 1

This convention is important in combinations and probability.

Find 7!

Formula: 7! = 7 × 6!

Substitution: 7 × 720

Answer: 5,040

Factorials can be built from smaller factorials.

Permutation example

Formula: nPr = n! ÷ (n − r)!

Substitution: 5P3 = 5! ÷ 2!

Answer: 60

Factorials help count ordered arrangements.

Combination example

Formula: nCr = n! ÷ (r! × (n − r)!)

Substitution: 5C2 = 5! ÷ (2! × 3!)

Answer: 10

Combinations divide out repeated orderings.

Fast growth example

Formula: 10!

Substitution: 10 × 9 × ... × 1

Answer: 3,628,800

Even small factorials can become large quickly.

Factorials in Counting and Probability

Arrangements

Factorials count ways to arrange distinct items.

Permutations

Use factorials when order matters.

Combinations

Use factorials and divide duplicate orderings when order does not matter.

Probability

Factorials help count possible outcomes.

Statistics

Factorials appear in binomial and distribution formulas.

Series

Factorials appear in Taylor series and advanced math.

Common Factorial Mistakes

Trying to calculate factorials for negative numbers.
Using decimals when a standard factorial needs a whole number.
Forgetting that 0! equals 1.
Confusing n! with n × n.
Underestimating how quickly factorials grow.
Using factorials when order does not matter without dividing duplicate arrangements.
Rounding or shortening large factorials too early.
Mixing permutations and combinations.

Understanding Your Result

Factorial result

The exact product of all positive whole numbers from n down to 1.

Expanded form

Shows the multiplication chain for small values of n.

Digit count

Shows how large the factorial result is.

Compact result

Shortens very large values so they stay readable on mobile.

Counting meaning

Represents the number of ways to arrange n distinct items.

Frequently Asked Questions

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