HomeCalculatorsMathNumber Theory & Discrete MathPermutation Combination Calculator
Math calculator

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 CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes
Input interpretation
Enter values to calculate.
Result
—
Verified scope
all published modes
Assurance
Core
Declared partition coverage
PASS · 3/3 declared partitions (combination, permutation, invalid-domain) · Matrix
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Exact P(n,k) or C(n,k) for integers 0 ≤ k ≤ n ≤ 1000.
Scope
Integers 0 ≤ k ≤ n ≤ 1000
Verification
Engine tested · Specification checked · v1.0.0 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-22.discrete-algebra-cvp
Verification revision
2026-09-22.discrete-algebra-cvp · 3/3 property · digest 423bc06f604d
Legacy regression
11/11 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
5/5 golden · 4/4 CVP boundary · 7/7 invalid · 3/3 property · 3/3 cross-interface · 2/2 cross-calculator · 6/6 CVP contract · Manifest
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 legacy golden · 3 legacy boundary · legacy regression suite · 5/5 oracle-backed golden · 7/7 invalid · Artifact integrity PASS
This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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 6/6 · 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: Arithmetic Sequence Calculator, Factorial Calculator, Fibonacci Calculator, GCF Calculator, Geometric Sequence Calculator, LCM Calculator, Modular Inverse Calculator, Modulo Calculator . Explore all Number Theory & Discrete Math.

→ Arithmetic Sequence Calculator Nth term and partial sum of an integer arithmetic sequence. Same engine as Sequence. Not geometric. Runs locally. › → Factorial Calculator Calculate n! for integers 0 ≤ n ≤ 5000. Fast local exact integers with shareable, machine-readable results. › → Fibonacci Calculator Fibonacci numbers F_n for n ≤ 1000. F_0=0, F_1=1. Same engine as Sequence. Exact BigInt. Runs locally. › → 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. › → Geometric Sequence Calculator Nth term and partial sum of an integer geometric sequence. Same engine as Sequence. Not arithmetic. Runs locally. › → LCM Calculator Find the LCM of two or more integers with exact arithmetic and independent verification. Free, no sign-up. › → Modular Inverse Calculator Modular inverse of a modulo m when gcd(a,m)=1. Same engine as Euclidean remainder. Not a second GCF page. Runs locally. › → Modulo Calculator Integer remainder with a = q·m + r and 0 ≤ r < |m|. Euclidean convention, not JavaScript truncated %. ›

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.