Math

Factorial Calculator (n!) with Reference Table

Q: What is Stirling's approximation and when should I use it?

Stirling's approximation is **n! ≈ √(2πn) × (n/e)^n**. Use it when n is large and an exact value is impractical to compute by hand — for example in statistical mechanics, information theory, and asymptotic algorithm analysis. Its relative error drops below 1% for n ≥ 3 and below 0.1% for n ≥ 10.

Q: How many digits does 100! have?

**100! has exactly 158 digits** and starts with 9.332 × 10^157. Estimate the digit count using Stirling: log₁₀(n!) ≈ n·log₁₀(n/e) + 0.5·log₁₀(2πn). For n=100 this gives ≈ 157.97, so 158 digits. 100! also ends in exactly **24 trailing zeros**.

Q: How many trailing zeros does n! have?

Count trailing zeros using Legendre's formula: **⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …** (sum of integer divisions by powers of 5). For n=100: 20 + 4 = **24 zeros**. For n=1000: 200 + 40 + 8 + 1 = **249 zeros**. Each factor of 5 paired with a factor of 2 creates one trailing zero; factors of 2 always outnumber factors of 5.

Q: What real-world applications use factorials?

Factorials appear in: **Probability** (binomial and Poisson distributions), **Statistical mechanics** (Boltzmann entropy formula uses N! for particle microstates), **Computer science** (the Travelling Salesman Problem for n=20 cities has 19! ≈ 1.2 × 10^17 routes), **Genomics** (counting possible gene orderings), and **Cryptography** (permutation cipher key spaces — for a 256-element permutation, 256! ≈ 8.58 × 10^506).

Calculate n! instantly. Enter any non-negative integer and get the exact factorial value with a reference table from 0! to 20!, worked examples for permutations and combinations, and Stirling's approximation.

Data verified · June 2026
Edited by Martín Rodríguez
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Edited by: Martín Rodríguez (editorial policy )

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The factorial of n, written n!, is the product of every positive integer from 1 up to n. It answers the question: In how many ways can n distinct objects be arranged in order? For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow explosively fast — 20! already exceeds 2.4 quintillion — making this calculator essential for combinatorics, probability, statistics, and computer science problems where exact large-integer results matter.

When to use this calculator

Counting the number of ways to arrange 8 competitors on a podium: 8! = 40,320 possible orderings.
Calculating permutations of a 13-card hand dealt from a shuffled deck using 52! / 39!.
Computing binomial coefficients C(n, k) = n! / (k! × (n−k)!) needed for probability distributions and statistical tests.
Determining the number of unique routes a delivery driver can take visiting 10 stops: (10−1)! = 362,880 circular permutations.
Evaluating Taylor/Maclaurin series terms such as e^x = Σ (x^n / n!) in calculus and physics simulations.
Checking hash collision probabilities using the birthday problem approximation involving k! terms.

Factorial Values: n! from 0 to 20

n	n!	Scientific Notation	Number of Digits
0	1	1 × 10⁰	1
1	1	1 × 10⁰	1
2	2	2 × 10⁰	1
3	6	6 × 10⁰	1
4	24	2.4 × 10¹	2
5	120	1.2 × 10²	3
6	720	7.2 × 10²	3
7	5,040	5.04 × 10³	4
8	40,320	4.032 × 10⁴	5
9	362,880	3.629 × 10⁵	6
10	3,628,800	3.629 × 10⁶	7
11	39,916,800	3.992 × 10⁷	8
12	479,001,600	4.79 × 10⁸	9
13	6,227,020,800	6.227 × 10⁹	10
14	87,178,291,200	8.718 × 10¹⁰	11
15	1,307,674,368,000	1.308 × 10¹²	13
16	20,922,789,888,000	2.092 × 10¹³	14
17	355,687,428,096,000	3.557 × 10¹⁴	15
18	6,402,373,705,728,000	6.402 × 10¹⁵	16
19	121,645,100,408,832,000	1.216 × 10¹⁷	18
20	2,432,902,008,176,640,000	2.433 × 10¹⁸	19

Fuente: NIST Digital Library of Mathematical Functions §5.1–5.2 (dlmf.nist.gov/5)

How it works

How It Is Calculated

The factorial function is defined both recursively and iteratively:

Iterative:  n! = 1 × 2 × 3 × … × n
Recursive:  n! = n × (n−1)!  (base case: 0! = 1)

Stirling's Approximation (large n):
  n! ≈ √(2πn) × (n/e)^n
  where e ≈ 2.71828

The convention 0! = 1 ensures combinatorial formulas remain consistent when k = 0 or k = n.

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Reference Table: Factorial Values 0 to 20

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Large Factorial Quick Reference

n	n! (approx.)	Digits
25	1.551 × 10²⁵	26
50	3.041 × 10⁶⁴	65
100	9.333 × 10¹⁵⁷	158
170	7.257 × 10³⁰⁶	307
171	overflow	exceeds double-precision floating point

> Number of digits in n! ≈ n·log₁₀(n/e) + ½·log₁₀(2πn) — for n = 1,000 that is 2,568 digits.

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Worked Examples

Permutations — arranging a playlist of 7 songs in every possible order:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 unique orderings.

Combinations — choosing a 5-person committee from 10 candidates:
C(10, 5) = 10! / (5! × 5!) = 3,628,800 / (120 × 120) = 252 distinct committees.

Probability — how unique is a shuffled deck of 52 cards?
Total distinct orderings: 52! ≈ 8.066 × 10⁶⁷ — essentially every fair shuffle in history is a unique event.

Taylor series — computing e¹:
e¹ = 1/0! + 1/1! + 1/2! + 1/3! + … = 1 + 1 + 0.5 + 0.1667 + … ≈ 2.71828

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Common Mistakes

1. Assuming 0! = 0. The factorial of zero is 1 — required for consistent combinatorial formulas.
2. Applying factorials to negative integers. n! is undefined for negative integers in the standard definition.
3. Applying factorials to non-integers. 3.5! requires the Gamma function: Γ(4.5) ≈ 11.632.
4. Integer overflow in code. C (32-bit int) overflows at 13!. Use 64-bit integers up to 20!, or arbitrary-precision for larger values.
5. Confusing n! with n^n. 5! = 120 but 5^5 = 3,125.

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Example: 6!

n = 6

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

6! = 720

Frequently asked questions

Why does 0! equal 1 and not 0?

0! = 1 is a definition required for combinatorial formulas to remain consistent. The number of ways to arrange zero objects is exactly 1 (the empty arrangement). Without this convention, C(n, 0) = n!/(0! × n!) would be undefined instead of correctly equaling 1.

What are the factorial values from 1 to 10?

1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720, 7! = 5,040, 8! = 40,320, 9! = 362,880, 10! = 3,628,800. Each value equals the previous one multiplied by n: 6! = 6 × 5! = 6 × 120 = 720.

What is the largest factorial a standard calculator can compute exactly?

A 64-bit integer (uint64) holds values up to 2^64 − 1 ≈ 1.84 × 10^19, accommodating up to 20! (≈ 2.43 × 10^18) exactly but overflowing at 21!. This calculator uses double-precision floating point and works accurately up to 170! (≈ 7.26 × 10^306); 171! exceeds the double-precision maximum.

How fast do factorials grow compared to exponential functions?

Factorials grow faster than any fixed exponential base. For large n, n! grows faster than 2^n, 10^n, or even 1,000^n. Stirling's approximation shows n! ≈ √(2πn)(n/e)^n — a super-exponential growth rate. For comparison, 20! = 2.43 × 10^18 while 2^20 = only 1,048,576.

What is Stirling's approximation and when should I use it?

Stirling's approximation is n! ≈ √(2πn) × (n/e)^n. Use it when n is large and an exact value is impractical to compute by hand — for example in statistical mechanics, information theory, and asymptotic algorithm analysis. Its relative error drops below 1% for n ≥ 3 and below 0.1% for n ≥ 10.

Can I compute the factorial of a decimal like 2.5!?

Not with the standard n! definition, which requires non-negative integers. For non-integer values, use the Gamma function: Γ(n+1) = n!, so 2.5! = Γ(3.5) ≈ 3.3234. The Gamma function is available in Python as math.gamma(3.5), in R as gamma(3.5), and in Wolfram Alpha.

How many digits does 100! have?

100! has exactly 158 digits and starts with 9.332 × 10^157. Estimate the digit count using Stirling: log₁₀(n!) ≈ n·log₁₀(n/e) + 0.5·log₁₀(2πn). For n=100 this gives ≈ 157.97, so 158 digits. 100! also ends in exactly 24 trailing zeros.

How many trailing zeros does n! have?

Count trailing zeros using Legendre's formula: ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … (sum of integer divisions by powers of 5). For n=100: 20 + 4 = 24 zeros. For n=1000: 200 + 40 + 8 + 1 = 249 zeros. Each factor of 5 paired with a factor of 2 creates one trailing zero; factors of 2 always outnumber factors of 5.

What real-world applications use factorials?

Factorials appear in: Probability (binomial and Poisson distributions), Statistical mechanics (Boltzmann entropy formula uses N! for particle microstates), Computer science (the Travelling Salesman Problem for n=20 cities has 19! ≈ 1.2 × 10^17 routes), Genomics (counting possible gene orderings), and Cryptography (permutation cipher key spaces — for a 256-element permutation, 256! ≈ 8.58 × 10^506).

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Editorial

Math calculator with its formula verified automatically against NIST Digital Library of Mathematical Functions — Gamma Function and Factorials (§5.1–5.2), per our editorial policy and methodology.

Updates

Updated: June 2026. Parameters are verified periodically against the cited sources.

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📌 How to cite this calculator

APA format

Rodríguez, M. (2026). Factorial Calculator (n!) with Reference Table. Hacé Cuentas. https://hacecuentas.com/en/factorial-numero-n

BibTeX

@misc{hacecuentas_factorial_numero_n_2026,
  author       = {Rodríguez, Martín},
  title        = {{Factorial Calculator (n!) with Reference Table}},
  year         = {2026},
  howpublished = {\url{https://hacecuentas.com/en/factorial-numero-n}},
  note         = {Hacé Cuentas}
}

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