Math

Factorial, Combinations, and Permutations

Q: Why is 0! = 1?

By **mathematical convention** that keeps formulas consistent. `C(n, 0)` must be 1 (there's only one way to 'choose nothing'). For `C(n,0) = n!/(0!×n!)` to equal 1, `0!` must be 1. It also fits the gamma function: `Γ(1) = 1`, and `Γ(n+1) = n!`. Think of 0! as an **empty product** = 1 (analogous to empty sum = 0).

Q: What about factorial of negative numbers?

**Undefined** for negative integers (division by zero in the gamma extension). But the **gamma function** `Γ(n+1)` extends factorial to **reals and complexes**, except at negative integers 0, −1, −2, … where there are poles. `Γ(0.5) = √π ≈ 1.77`, `Γ(1.5) = 0.5 × √π ≈ 0.89`.

Q: When do I use permutations vs. combinations?

If **order matters** (podium, passwords, numbered seats), use **permutations** `P(n,r)`. If order **doesn't matter** (card hands, teams, groups), use **combinations** `C(n,r)`. Quick rule: ask 'does the result change if I change the order?' If yes, permutation. If no, combination.

Q: What is a binomial coefficient?

Another name for **combinations**. `C(n, r)` is also written `(n r)` ('n choose r') or `ⁿCᵣ`. It appears in the **binomial theorem**: `(a + b)ⁿ = Σ C(n, k) × aⁿ⁻ᵏ × bᵏ`. It forms **Pascal's triangle**, where each number is the sum of the two above.

Q: What is Pascal's triangle?

Triangular arrangement where each row `n` contains the binomial coefficients `C(n,0), C(n,1), …, C(n,n)`. Each number is the sum of the two directly above. Row 0: `1`. Row 1: `1 1`. Row 2: `1 2 1`. Row 3: `1 3 3 1`. Row 4: `1 4 6 4 1`. These numbers appear in the binomial theorem and combinatorics.

Q: How do I handle huge factorials without losing precision?

Use **BigInt** (JavaScript: `BigInt(n)` instead of `Number(n)`). Number loses precision beyond `2^53 ≈ 9 × 10¹⁵`. To compute `50!` or `100!`, you need BigInt. This calculator uses BigInt so you get exact factorials with hundreds of digits. For fast approximations, use **Stirling's formula**: `n! ≈ √(2πn) × (n/e)^n`.

Q: What if I have repeated elements?

They're **permutations with repetition**. If you have `n` total elements with `n₁` of type A, `n₂` of type B, etc., distinct permutations are `n! / (n₁! × n₂! × … × nₖ!)`. Example: word 'PAPA' has 4 letters (2 P, 2 A), distinct permutations: `4! / (2! × 2!) = 24/4 = 6`. Valid for 'AAPP', 'APAP', 'APPA', 'PAAP', 'PAPA', 'PPAA'.

Factorial calculator: compute n!, combinations C(n,r), and permutations P(n,r). Great for statistics, probability, combinatorics, and lottery. With BigInt.

Data verified · June 2026
Edited by Martín Rodríguez
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How to use this calculator

Follow this tool’s steps, then review its formula, assumptions, and limits below.

Step by step

Select what to calculateSelect from the dropdown: Factorial n!, Permutation P(n,r), or Combination C(n,r).

Enter nEnter n (e.g., 5).

Enter rEnter r (only for permutation/combination) (e.g., 3).

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n. Factorials, permutations, and combinations are foundational topics in US AP Statistics, college discrete math, and the AMS standard combinatorics curriculum. This calculator handles n!, P(n, r) (order matters) and C(n, r) (order doesn't matter), uses native BigInt so 100! (≈ 9.3 × 10¹⁵⁷) computes exactly, and is useful for Powerball/Mega Millions probability, poker hands, and password entropy.

When to use this calculator

You're calculating the probability of winning the lottery: C(n, r).
You want to know how many poker hands exist in a 52-card deck: C(52,5).
You're working on combinatorics problems in high school or college.
You're analyzing how many possible orderings a password or array has.
You're computing binomial coefficients for a binomial distribution in statistics.

Factorial Reference Values (n!)

n	n! (exact or approximation)
0	1
1	1
2	2
3	6
4	24
5	120
10	3,628,800
15	≈ 1.3 × 10¹²
20	2,432,902,008,176,640,000
30	≈ 2.7 × 10³²
50	≈ 3.0 × 10⁶⁴
70	≈ 1.2 × 10¹⁰⁰ (≈ 1 googol)
100	≈ 9.3 × 10¹⁵⁷
170	≈ 7.3 × 10³⁰⁶ (JavaScript Number limit)

Fuente: NIST Digital Library of Mathematical Functions — dlmf.nist.gov/5; valores de n=5, 10, 20, 100 verificados contra el contenido de la calculadora.

How it works

Factorial (n!)

n! = n × (n−1) × (n−2) × ... × 2 × 1
0! = 1 (by convention)
1! = 1

Examples:

5! = 5 × 4 × 3 × 2 × 1 = 120

10! = 3,628,800

20! = 2,432,902,008,176,640,000

70! ≈ 1.2 × 10¹⁰⁰ (roughly a googol)

100! ≈ 9.3 × 10¹⁵⁷

What It Means

Factorial counts the ways to arrange n distinct objects. If you have 5 different books on a shelf, there are 120 different ways to order them.

Permutations — P(n, r)

P(n, r) = n! / (n − r)!
        = n × (n−1) × (n−2) × ... × (n−r+1)

Ways to order r elements taken from n, where order matters.

Classic Example

How many ways to assign gold, silver, and bronze in a race of 10 runners?

P(10, 3) = 10! / 7! = 10 × 9 × 8 = 720

Order matters because Runner A in gold and Runner B in silver is different from Runner B in gold and Runner A in silver.

Combinations — C(n, r)

C(n, r) = n! / (r! × (n−r)!)

Ways to pick r elements out of n, regardless of order. Read as 'n choose r.' Also written ⁿCᵣ or as the binomial coefficient (n r).

Classic Example

How many 5-card hands exist in a 52-card deck?

C(52, 5) = 52! / (5! × 47!) = 2,598,960

Order doesn't matter because the hand is the set of cards, not their sequence.

Key Difference: Permutation vs. Combination

	Permutation	Combination
Order	Matters	Doesn't matter
Formula	`n! / (n−r)!`	`n! / (r! × (n−r)!)`
Example	Race podium	Card hand
Relationship	`P = C × r!`	`C = P / r!`
Always	`P(n,r) ≥ C(n,r)`

Mnemonic

If you can distinguish order (1st, 2nd, 3rd), it's a permutation.

If only which elements you picked matters, it's a combination.

Real-World Applications

Lottery Probability

Powerball-style (5 of 69 + 1 of 26): massively low odds.

Poker

Hand	Combinations	Probability
Royal flush	4	0.000154%
Straight flush	36	0.00139%
Four of a kind	624	0.024%
Full house	3,744	0.144%
Flush	5,108	0.197%
Any hand	2,598,960	100%

Combinatorics in Statistics

Binomial distribution: probability of getting k successes in n tries (flipping n coins).

P(X = k) = C(n, k) × p^k × (1−p)^(n−k)

Where p is the probability of success in one trial.

Algorithms and Big O

Traveling Salesman Problem (TSP): n! possible routes — explodes for n > 10.

Sorting: there are n! possible orderings of n elements.

Sudoku: ≈ 6.67 × 10²¹ valid boards.

Chess: more than 10⁴³ legal positions (Shannon's estimate).

Large Factorials

Factorials grow so fast they lose intuitive scale:

n	n!
5	120
10	≈ 3.6 × 10⁶
15	≈ 1.3 × 10¹²
20	≈ 2.4 × 10¹⁸
30	≈ 2.7 × 10³²
50	≈ 3.0 × 10⁶⁴
70	≈ 1.2 × 10¹⁰⁰ (a googol)
100	≈ 9.3 × 10¹⁵⁷
170	≈ 7.3 × 10³⁰⁶ (JavaScript Number limit)

Beyond that, BigInt is needed (arbitrarily large integers). This calculator uses BigInt for n up to 1000.

Gamma Function — extended factorial

The gamma function Γ(n+1) = n! extends factorial to real and complex numbers (except 0, −1, −2, …). Example: Γ(0.5) = √π. Used in advanced statistics (gamma, chi-squared, Student's t distributions).

Stirling's Approximation

For very large n, factorial is approximated with:

n! ≈ √(2πn) × (n/e)^n

Excellent approximation starting from n = 10. Useful in asymptotic analysis.

Common Mistakes

1. Computing 0!: many say 0. It's 1 by convention (so formulas keep working).
2. Confusing P(n,r) with C(n,r): ask whether order matters.
3. Brute-forcing C(52,5): don't compute 52!/47!/5!. Simplify: (52×51×50×49×48)/(5×4×3×2×1).
4. Permutations with repetition: if elements repeat (like letters in 'MISSISSIPPI'), the formula changes: n! / (n₁! × n₂! × ...).
5. Trying n! with JavaScript Number: overflow beyond n = 170. Use BigInt.

Example: How many 5-card hands can be formed from a 52-card deck?

Context: we pick 5 cards out of 52, order doesn't matter (3♠ 4♠ 5♠ 6♠ 7♠ is the same hand as 7♠ 6♠ 5♠ 4♠ 3♠).

Use combinations: C(52, 5) = 52! / (5! × 47!).

Simplification: (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1).

Numerator: 52 × 51 × 50 × 49 × 48 = 311,875,200.

Denominator: 5! = 120.

Result: 311,875,200 / 120 = 2,598,960.

There are C(52, 5) = 2,598,960 distinct 5-card poker hands in a standard US 52-card deck. The probability of drawing any one specific hand is 1 / 2,598,960 ≈ 0.0000385% (per NIST/SEMATECH combinatorics conventions).

Frequently asked questions

Why is 0! = 1?

By mathematical convention that keeps formulas consistent. C(n, 0) must be 1 (there's only one way to 'choose nothing'). For C(n,0) = n!/(0!×n!) to equal 1, 0! must be 1. It also fits the gamma function: Γ(1) = 1, and Γ(n+1) = n!. Think of 0! as an empty product = 1 (analogous to empty sum = 0).

What about factorial of negative numbers?

Undefined for negative integers (division by zero in the gamma extension). But the gamma function Γ(n+1) extends factorial to reals and complexes, except at negative integers 0, −1, −2, … where there are poles. Γ(0.5) = √π ≈ 1.77, Γ(1.5) = 0.5 × √π ≈ 0.89.

When do I use permutations vs. combinations?

If order matters (podium, passwords, numbered seats), use permutations P(n,r). If order doesn't matter (card hands, teams, groups), use combinations C(n,r). Quick rule: ask 'does the result change if I change the order?' If yes, permutation. If no, combination.

How do I calculate my Powerball odds?

Powerball: pick 5 of 69 white balls + 1 of 26 red. Combinations: C(69,5) × 26 = 292,201,338. Probability of jackpot: 1 / 292,201,338 ≈ 0.00000034%. You're more likely to be struck by lightning than win.

What is a binomial coefficient?

Another name for combinations. C(n, r) is also written (n r) ('n choose r') or ⁿCᵣ. It appears in the binomial theorem: (a + b)ⁿ = Σ C(n, k) × aⁿ⁻ᵏ × bᵏ. It forms Pascal's triangle, where each number is the sum of the two above.

What is Pascal's triangle?

Triangular arrangement where each row n contains the binomial coefficients C(n,0), C(n,1), …, C(n,n). Each number is the sum of the two directly above. Row 0: 1. Row 1: 1 1. Row 2: 1 2 1. Row 3: 1 3 3 1. Row 4: 1 4 6 4 1. These numbers appear in the binomial theorem and combinatorics.

How do I handle huge factorials without losing precision?

Use BigInt (JavaScript: BigInt(n) instead of Number(n)). Number loses precision beyond 2^53 ≈ 9 × 10¹⁵. To compute 50! or 100!, you need BigInt. This calculator uses BigInt so you get exact factorials with hundreds of digits. For fast approximations, use Stirling's formula: n! ≈ √(2πn) × (n/e)^n.

What if I have repeated elements?

They're permutations with repetition. If you have n total elements with n₁ of type A, n₂ of type B, etc., distinct permutations are n! / (n₁! × n₂! × … × nₖ!). Example: word 'PAPA' has 4 letters (2 P, 2 A), distinct permutations: 4! / (2! × 2!) = 24/4 = 6. Valid for 'AAPP', 'APAP', 'APPA', 'PAAP', 'PAPA', 'PPAA'.

How many distinct passwords exist?

Depends on alphabet and length. 8-character password with lowercase letters (26 options): 26^8 = 208,827,064,576 (~209 billion). Adding uppercase (52): 52^8 ≈ 5.35 × 10¹³. With digits (62): 62^8 ≈ 2.18 × 10¹⁴. With symbols (94): 94^8 ≈ 6.10 × 10¹⁵. That's why length matters more than complexity for password security.

Sources & references

Methodology & trust

Editorial

Math calculator with its formula verified automatically against NIST Digital Library of Mathematical Functions — Factorials and Combinatorial Identities, per our editorial policy and methodology.

Updates

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

Privacy

Calculations run 100% in your browser. We do not store or transmit your data.

Limitations

Indicative results. For critical decisions, consult a professional.

📌 How to cite this calculator

APA format

Rodríguez, M. (2026). Factorial, Combinations, and Permutations. Hacé Cuentas. https://hacecuentas.com/en/factorial-combinations-permutations

BibTeX

@misc{hacecuentas_factorial_combinations_permutations_2026,
  author       = {Rodríguez, Martín},
  title        = {{Factorial, Combinations, and Permutations}},
  year         = {2026},
  howpublished = {\url{https://hacecuentas.com/en/factorial-combinations-permutations}},
  note         = {Hacé Cuentas}
}

Content licensed under CC-BY 4.0 — reuse it citing the source with a link to Hacé Cuentas.

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Factorial, Combinations, and Permutations 🌎

How to use this calculator

When to use this calculator

Factorial Reference Values (n!)

How it works

Factorial (n!)

What It Means

Permutations — P(n, r)

Classic Example

Combinations — C(n, r)

Classic Example

Key Difference: Permutation vs. Combination

Mnemonic

Real-World Applications

Lottery Probability

Poker

Combinatorics in Statistics

Algorithms and Big O

Large Factorials

Gamma Function — extended factorial

Stirling's Approximation

Common Mistakes

Example: How many 5-card hands can be formed from a 52-card deck?

Frequently asked questions

Sources & references

Methodology & trust

Related calculators

Factorial, Combinations, and Permutations