It comes from definition and from consistency with combinatorics. C(n,0) = 1 because there is exactly one way to choose nothing. The empty product is 1, the multiplicative identity. The recursion (n+1)! = (n+1) × n! requires 0! to be 1 so that 1! = 1 × 0! = 1.

How fast does factorial grow?

Extremely fast. 5! is 120, 10! is 3.6 million, 15! is 1.3 trillion, 20! is 2.4 quintillion, 50! has 65 digits, and 100! has 158 digits. Factorial grows faster than exponential growth, which limits computation precision.

What's a factorial used for?

Counting permutations — there are n! ways to arrange n items. Combinations are C(n,r) = n!/(r!(n-r)!). Factorials show up in probability calculations, calculus series like Taylor expansions, and computer science topics like recursion examples and complexity analysis.

Can I calculate factorials of decimals?

Not the standard factorial, which is defined only for positive integers. The Gamma function generalizes it, where Γ(n+1) = n!, and that lets you take the factorial of a decimal — 0.5! works out to about 0.886. This is beyond the scope of a basic factorial calculator.

What about very large factorials?

Large factorials are a computational challenge. 1000! has roughly 2570 digits. Python and JavaScript with BigInt can compute them; standard floating point overflows around 170!. For approximation, Stirling's formula gives n! ≈ √(2πn)(n/e)^n.

What's Stirling's approximation?

Stirling's approximation says n! ≈ √(2πn) × (n/e)^n, and it is accurate for n greater than 10. It shines when an exact value is impractical (very large) or unnecessary (you just need the order of magnitude), and it appears in physics, statistics, and complexity analysis.

Is the data sent anywhere?

No. Factorial calculation is pure JavaScript math, running entirely in your browser. Your input never leaves your device.

Factorial Calculator

Calculate the factorial of any number online with support for large values. Free factorial calculator for math and combinatorics work.

Calculators

Instant results

Number (0-1000)

10! =

3628800

7 digits

How to Use Factorial Calculator

Enter a number

Type any positive integer or zero. The tool calculates n! immediately.

View result

The output shows the full factorial value, scientific notation for large values, and optionally the intermediate steps (such as 5×4×3×2×1=120).

Compare with related

The tool may also show the 0! through n! sequence, a growth comparison, and related calculations like combinations and permutations.

Use in math problems

Apply the results to probability calculations, combinatorics, programming verification, and mathematical homework.

When to Use Factorial Calculator

Mathematical computations and combinatorics

Combinatorics, probability, and permutations all rely heavily on factorials. The tool computes n! quickly for any non-negative integer n, which fits homework problems, prepared classroom examples, calculations of combinations and permutations in research, partition function work in physics, and statistical analysis of sample arrangements.

Programming and algorithm verification

Comparing your code's output against known factorial values is a fast way to verify an implementation. The tool helps with testing recursive against iterative implementations, debugging factorial-based algorithms, exploring algorithmic complexity through factorial growth, validating combinatorial code in production, comparing BigInt and standard integer overflow behavior, and providing reference values for computer science coursework. Edge cases like 0!, 1!, and very large factorials are worth testing explicitly.

Probability and statistics calculations

The permutations formula P(n,r) = n!/(n-r)! and the combinations formula C(n,r) = n!/(r!(n-r)!) both depend on factorials. The tool produces accurate n! values that feed into these formulas, which is useful in statistics coursework, gambling odds, decision theory, lottery probability, card game mathematics, dice combination analysis, and anything that involves counting arrangements or selections from sets.

Educational learning and mathematical exploration

Factorials grow at a startling rate. 5! is just 120, but 10! is already 3.6 million, 15! climbs over a trillion, and 20! exceeds 2.4 quintillion. The tool reveals this growth interactively, which helps students learn combinatorics, visualize rapid-growth functions, understand why factorial complexity is impractical for large n in algorithms, and grasp how quickly counting problems run past computational feasibility.

Factorial Calculator Examples

Simple factorial

Input

5!

Output

5! = 5 × 4 × 3 × 2 × 1 = 120

A basic factorial: 5 × 4 × 3 × 2 × 1 = 120. This pattern shows up in probability problems, permutation counting, and basic combinatorics.

Larger factorial

Input

20!

Output

20! = 2,432,902,008,176,640,000 (~2.43 × 10^18)

20 factorial reaches 2.4 quintillion, which approaches the limits of 64-bit integer arithmetic. Anything larger requires BigInt support, which the tool typically provides.

Stirling approximation

Input

100! approximated

Output

Exact: huge number (158 digits). Stirling: ~9.33 × 10^157.

Stirling's approximation says n! ≈ √(2πn)(n/e)^n, and it shines for very large factorials where exact computation is impractical. It comes up regularly in physics, statistics, and complexity analysis.

Tips & Best Practices for Factorial Calculator

1.Factorial grows extremely fast. 10! is 3.6 million, 20! is 2.4 quintillion, and 100! has 158 digits. Watch for overflow in your implementation.
2.Use BigInt when the inputs get large. Standard 32-bit and 64-bit integers overflow around 12! and 20! respectively, while JavaScript's BigInt handles arbitrary size.
3.0! equals 1 by definition. This is easy to forget, and it shows up naturally in combinations where C(n,0) = 1.
4.Negative factorials are undefined. Real-valued factorials extend through the Gamma function, where Γ(n+1) = n!, but that goes beyond the scope of a basic factorial calculator.
5.When you actually need probability calculations, look for combinations or permutations tools directly — they tend to be more useful than raw factorial.
6.Stirling's approximation, n! ≈ √(2πn)(n/e)^n, is accurate for large n and useful whenever an exact value is impractical to compute.

Frequently Asked Questions

A factorial is the product of all positive integers up to a given number, written n!. So n! = n × (n-1) × ... × 1. Examples: 5! is 5×4×3×2×1 = 120, 0! is defined as 1, and 1! is 1. Factorials show up in probability, combinatorics, calculus, and computer science.

Factorial Calculator

How to Use Factorial Calculator

Enter a number

View result

Compare with related

Use in math problems

When to Use Factorial Calculator

Mathematical computations and combinatorics

Programming and algorithm verification

Probability and statistics calculations

Educational learning and mathematical exploration

Factorial Calculator Examples

Simple factorial

Larger factorial

Stirling approximation

Tips & Best Practices for Factorial Calculator

Frequently Asked Questions

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