Factorial: n! = n · (n−1) · ... · 1

The factorial of a positive integer n (written n!) is the product of all integers from 1 to n. Convention: 0! = 1. Explosive growth: 10! = 3,628,800, 20! ≈ 2.4 × 10¹⁸, 170! ≈ 7.3 × 10³⁰⁶ (double precision limit). Beyond, use Gamma function or BigInt. This calculator has 5 modes: integer factorial, double factorial n!!, subfactorial !n, Gamma Γ(x) for reals, Stirling approximation.

Fundamental formulas

• n! = n · (n−1)! (recursive) with 0! = 1
• n! = ∏ᵢ₌₁ⁿ i
• Γ(n) = (n−1)! for positive integers
• Stirling: n! ≈ √(2πn) · (n/e)ⁿ
• Double: n!! = n·(n−2)·(n−4)·...
• Subfactorial: !n = n!·Σₖ (−1)ᵏ/k!

Common factorials

Examples

Example 1: permutations of 6

Arranging 6 people in a row: 6! = 720 ways.

Example 2: lottery combinations

Pick 6 from 90 (Superenalotto): C(90,6) = 622,614,630. Probability: 1 in 622M.

Example 3: card deck

52 cards shuffles: 52! ≈ 8.07 × 10⁶⁷. More than atoms on Earth (10⁵⁰).

Example 4: derangement

5 people hat swap: !5 = 44. ~37% of permutations are derangements.

Example 5: Stirling for n=20

Exact: 20! = 2,432,902,008,176,640,000. Stirling ~2.42×10¹⁸. Error ~0.42%.

Gamma function Γ(x)

Extension of factorial to reals/complexes:

• Γ(n+1) = n! for positive integers
• Γ(1/2) = √π ≈ 1.7725
• Γ(x+1) = x·Γ(x)
• Poles at 0, −1, −2, ...

Introduced by Euler 1729. Numeric via Lanczos (15 digits precision).

Stirling approximation (1730)

n! ≈ √(2πn) · (n/e)ⁿ

More accurate: ·(1 + 1/(12n) + 1/(288n²) − ...).

Error: ~1/(12n). n=10: 0.83%. n=100: 0.083%. Used in asymptotic analysis, statistical mechanics, number theory.

Double factorial n!!

Product of same-parity integers: n!! = n·(n−2)·(n−4)·...

Examples: 5!! = 5·3·1 = 15. 6!! = 48. 10!! = 3840.

Relations: (2n)!! = 2ⁿ·n!. Used in combinatorics (Dyck paths), physics (Gauss moment integrals).

Subfactorial !n

Counts permutations without fixed points (derangements):

!n = n!·Σₖ (−1)ᵏ/k!

Values: !1=0, !2=1, !3=2, !4=9, !5=44, !6=265.

lim !n/n! = 1/e ≈ 0.3679. 37% of any random permutation is a derangement.

Hat problem

N people deposit hats at entrance. Exit randomly. Probability nobody gets their own = !N/N! → 1/e as N→∞. 36.79% at infinity.

Permutations and combinations

• P(n,k) = n!/(n−k)! — ordered selections
• C(n,k) = n!/(k!·(n−k)!) — unordered selections
• Binomial (ⁿₖ) = C(n,k)

Example: lottery 6/49 → C(49,6) = 13,983,816 possibilities.

Applications

Combinatorics and probability

• Lotteries (SuperEnalotto, Powerball, EuroMillions)
• Poker: C(52,5) = 2,598,960 hands
• Passwords
• Graph paths
• Anagrams

Statistics

• Poisson: P(k) = e^(−λ)·λᵏ/k!
• Binomial: C(n,k)·pᵏ·(1−p)^(n−k)
• Chi-square

Physics and chemistry

• Statistical mechanics: S = k·log(Ω)
• Quantum mechanics
• Taylor series: e^x = Σ xⁿ/n!
• Special functions (Bessel, Legendre)

Computer science

• Recursive algorithms
• O(n!) complexity: traveling salesman brute force
• Permutation generation

Engineering

• Maxwell-Boltzmann distribution
• Statistical quality control
• System reliability

Taylor series with factorials

• e^x = Σ xⁿ/n!
• sin(x) = Σ (−1)ⁿ x^(2n+1)/(2n+1)!
• cos(x) = Σ (−1)ⁿ x^(2n)/(2n)!
• (1+x)^α = Σ C(α,n)·xⁿ

History

Clavius (1603): first use in recreational math.

Kramp (1808): introduces "n!" notation.

Euler (1729): extends to Gamma function for reals. Extraordinary feat: generalizing n! to all numbers.

Stirling (1730): asymptotic approximation.

Ramanujan (1914): extraordinary series for factorials.

Trivia

• Number of digits of n!: log₁₀(n!) ≈ log₁₀(√(2πn)) + n·log₁₀(n/e)
• Trailing zeros of n!: = number of 5s in factorization = Σⁱₖ₌₁ ⌊n/5ᵏ⌋
• Example: 25! has 5+1 = 6 trailing zeros

FAQ

Why 0! = 1?

Convention. Empty product = 1 (neutral element). Enables elegant formulas: C(n,0) = 1. Consistent with Γ(1) = 1.

Why does n! explode?

Multiplies all positive integers up to n. 10! = 3.6M, 20! is 18 digits, 100! has 158 digits. Grows faster than any exponential.

How to calculate large n!?

(

• Exact with BigInt; (
• Stirling approximation (~1% error); (
• Gamma via Lanczos (15 digits). For n>170 use BigInt (double overflow)

Gamma function?

Extension to reals/complexes: Γ(n) = (n−1)! for positive integers. Γ(1/2) = √π. Poles at 0, −1, −2. Fundamental in statistics, physics.

Difference factorial vs double factorial?

n! all integers down to 1. n!! same-parity only. Ex: 5!=120, 5!!=15.

Use of derangements?

Permutations without fixed points. Hat problem, secret santa, file matching, cryptography.

Fast computation of n!?

Stirling: n! ≈ √(2πn)(n/e)ⁿ. For n=100: 9.32×10¹⁵⁷ (exact 9.33). ~0.08% error. Good enough for most non-financial apps.

Common errors

• 0! = 0 (FALSE, it's 1)
• Factorial of non-integers (only Gamma)
• Silent overflow beyond 170 (Infinity in JS)
• Confusing n!! with (n!)!
• Negative factorials (undefined, Gamma poles)
• Floating point for exact combinatorics

Updated July 2026