Factorial (n!) Calculator

Factorial calculator: what it computes

The factorial of a non‑negative integer n, written n!, is the product of all whole numbers from 1 up to n. This page computes exact factorial values in your browser using BigInt, so results are not rounded (within practical performance limits).

Quick definition

For integers n ≥ 1:

n! = n × (n − 1) × (n − 2) × … × 2 × 1

And by convention:

0! = 1 (the “empty product”).

Formula: MathML (product form)

Recursive form (useful for reasoning)

Factorials satisfy:

n! = n × (n − 1)!, with 0! = 1.

How to use this factorial calculator

1. Enter a whole number n ≥ 0.
2. Select Compute Factorial (or press Enter).
3. Review the outputs: the exact value of n! and the number of digits.
4. Use Copy Result to copy the factorial value to your clipboard.

This tool is intentionally limited to standard factorials for non‑negative integers. If you need “factorials” of decimals or negative numbers, that’s typically the Gamma function, which is not computed here.

Factorials grow extremely fast. Even modest inputs produce very large integers:

• 10! = 3,628,800 (already millions)
• 20! is larger than what fits in a 64‑bit signed integer
• 100! has 158 digits

That rapid growth is exactly why factorials appear in counting and probability problems: they count how many ways to arrange or choose items.

Worked example: computing and reading 6!

Suppose you want to compute 6!.

By the definition:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Interpretation: if you have 6 distinct objects (for example, 6 different books), there are 720 different possible orderings.

The same number feeds directly into combinatorics. How many ways can you pick 2 of those 6 books to take on a trip, ignoring order? C(6, 2) = 6! ÷ (2! × 4!) = 720 ÷ (2 × 24) = 15. Enter 6 above and the calculator reports the exact value 720, a digit count of 3, and 1 trailing zero (from the single factor of 5 in the product 6 × 5 × 4 × 3 × 2 × 1). Enter 100 and it reports 158 digits, 24 trailing zeros, and an approximate scientific form of 9.332 × 10¹⁵⁷.

Common factorial values (and digit counts)

This table is helpful for sanity‑checking results and understanding growth.

Introduction: Why factorials matter (common uses)

Permutations (arrangements)

If you have n distinct items and want to count how many different orderings exist, the answer is n!.

Combinations (choosing without order)

The number of ways to choose r items from n items is:

C(n, r) = n! / (r!(n − r)!)

That’s why factorials show up constantly in probability, statistics, and counting problems.

Series expansions (why n! appears in calculus)

Factorials appear in power series; for example, the exponential function can be written as:

e^x = Σ (xⁿ / n!) for n = 0 to ∞.

Estimating size: Stirling’s approximation and digit counts

When you only need the magnitude of a factorial rather than its exact digits, Stirling’s approximation is the standard tool:

Already at n = 10 the approximation gives 3,598,696 against the true 3,628,800 — an error under 1 percent that keeps shrinking as n grows. Taking base-10 logarithms turns it into a digit-count predictor: the number of digits of n! is ⌊log₁₀(n!)⌋ + 1, and log₁₀(n!) ≈ n⋅log₁₀(n/e) + ½⋅log₁₀(2πn). For n = 100 this predicts 157.97, matching the true 158 digits. This is exactly how the calculator’s approximate scientific notation stays honest even when the exact integer runs to thousands of digits.

Why factorials end in zeros (Legendre’s formula)

Every trailing zero of n! is a factor of 10 = 2 × 5, and since factors of 2 are plentiful, the fives are the bottleneck. Legendre’s formula counts them:

For n = 100: ⌊100/5⌋ + ⌊100/25⌋ = 20 + 4 = 24 trailing zeros, which you can confirm in the result panel. The sum is finite in practice because terms vanish once 5ⁱ > n. Interviewers love this one, and it is a good sanity check that a huge factorial result is correct: wrong arithmetic almost never lands on exactly the right zero count.

Plain-text formula: n! = n × (n − 1) × … × 2 × 1 with 0! = 1; digits(n!) = floor(log10(n!)) + 1; Stirling: n! ≈ √(2πn) × (n/e)ⁿ; trailing zeros Z(n) = ∑ floor(n/5ⁱ) for i = 1, 2, …; scientific form shown as first digits × 10^(digits − 1).

Source/version metadata: exact values computed client-side with JavaScript BigInt (arbitrary-precision integers, no rounding); Stirling’s approximation and Legendre’s formula are classical results found in any combinatorics text (e.g., Graham, Knuth & Patashnik, Concrete Mathematics, ch. 4). Input capped at n = 5,000 for responsiveness. Last reviewed July 2026.

Limitations and assumptions (important)

• Integer‑only: This calculator accepts whole numbers only. Decimals (e.g., 2.5) and negatives are rejected because standard factorial is defined for integers n ≥ 0.
• Practical performance cap: Very large inputs produce extremely large integers. To keep the page responsive, the calculator enforces a maximum (see the form’s “Max” note). You can lower the input if your device slows down.
• Output can be enormous: Rendering or copying thousands of digits can be slow and may stress older browsers/devices.
• No approximations: The output is exact, not scientific notation. If you only need magnitude (e.g., number of digits), you may prefer approximation methods (e.g., Stirling’s approximation) in a dedicated tool.
• Browser support: Exact computation relies on BigInt, which is supported by modern browsers. In older environments, this calculator may not work.

Factorial questions students actually ask