HomeCalculatorsMathNumber Theory & Discrete MathPermutation Combination Calculator
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Permutation Combination Calculator

Exact P(n,k) and C(n,k) for integers 0 ≤ k ≤ n ≤ 1000. Multiplicative BigInt, local-first. Not the factorial calculator.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 11/11 tests · Production surface contract 4/4 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
Exact P(n,k) or C(n,k) for integers 0 ≤ k ≤ n ≤ 1000.
Scope
Integers 0 ≤ k ≤ n ≤ 1000
Verification
Engine tested · 11/11 tests · Production surface contract 4/4 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics (combinatorial numbers)
  • NIST / standard discrete-math identity P(n,k) = C(n,k)·k!
Specification basis
Evidence
2 golden · 3 boundary · 6 property · Production surface contract 4/4 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-16T06:44:24.909Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

PermutationP(n,k) = n! / (n − k)!
CombinationC(n,k) = n! / (k!(n − k)!)
IdentityP(n,k) = C(n,k) × k!
iThis page uses the multiplicative product, not a call to the factorial calculator. Domain 0 ≤ k ≤ n ≤ 1000. Multinomials and with-replacement counts are out of scope.

How to use

1

Choose permutation or combination

Permutation (nPr) is ordered. Combination / binomial (nCr) is unordered.

2

Enter integers n and k

0 ≤ k ≤ n ≤ 1000. Fractions are rejected.

3

Read the exact integer

Results are exact. C(n,k) = C(n, n−k).

Example calculations

Common configurations with formula and result.

ϟ

C(10,3)

Unordered selections of 3 from 10

C(10,3) = 10! / (3!·7!)
120
ϟ

P(10,3)

Ordered selections of 3 from 10

P(10,3) = 10×9×8
720
ϟ

Empty selection

C(n,0) = 1

C(8,0)
1

Permutation Combination calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
P(n,k) counts ordered selections of k items from n. C(n,k) counts unordered selections (binomial coefficient). Both require 0 ≤ k ≤ n.
What it calculates
Exact P(n,k) or C(n,k) for integers 0 ≤ k ≤ n ≤ 1000.
Inputs
  • mode?
  • n
  • k
Outputs
  • value
  • exact
  • digits
  • symbol
Formula
P(n,k)=n!/(n−k)!; C(n,k)=n!/(k!(n−k)!)
Assumptions
  • Integers 0 ≤ k ≤ n ≤ 1000
  • Not factorial n!
  • Not multinomial and not with-replacement
Units
  • dimensionless
Boundary conditions
  • missing n or k → MISSING_REQUIRED_INPUT
  • k > n → INVALID_INPUT
  • n > 1000 → VALUE_ABOVE_MAX
  • unknown mode → INVALID_MODE
Example
mode=combination n=10 k=3 → 120
Validation cases

3 published on this page · 11/11 tests · Production surface contract 4/4 · View evidence

  • mode=combination n=10 k=3 → 120
  • mode=permutation n=10 k=3 → 720
  • n=5 k=6 → error INVALID_INPUT
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics (combinatorial numbers)
  • NIST / standard discrete-math identity P(n,k) = C(n,k)·k!
Calculation version
1.0.0

Other calculators in this family: Factorial Calculator, GCF Calculator, LCM Calculator, Modulo Calculator, Prime Factorization Calculator . Explore all Number Theory & Discrete Math.

→ Factorial Calculator Calculate n! for integers 0 ≤ n ≤ 5000. Fast local exact integers with shareable, machine-readable results. › → GCF Calculator Find the GCF, GCD, or HCF of two or more integers. Exact BigInt Euclidean fold with factor and prime cross-checks. Free, no sign-up. › → LCM Calculator Find the LCM of two or more integers with exact arithmetic and independent verification. Free, no sign-up. › → Modulo Calculator Integer remainder with a = q·m + r and 0 ≤ r < |m|. Euclidean convention, not JavaScript truncated %. › → Prime Factorization Calculator Factor n = ∏ p^e and test primality for 1 ≤ n ≤ 10^12. Trial division. One capability, two modes. ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this the factorial calculator?

No. n! is /calc/math/factorial. This page computes P(n,k) and C(n,k). Internally it uses a multiplicative product, not n!/(n−k)! as a quotient of two factorial calls.

What is the difference between P and C?

P(n,k) cares about order. C(n,k) does not. P(n,k) = C(n,k)×k!.

Is C(n,k) the binomial coefficient?

Yes. Combination mode is C(n,k) = (n choose k).

Where does this run?

Locally in the browser by default. REST and MCP call the same combinatorics engine.