Permutation & Combination Calculator

Enter the total number of items (n) and the number to select (r). The calculator instantly computes P(n,r), C(n,r), and their repetition variants with step-by-step factorial breakdowns.

In permutations, order matters (e.g., lock codes, race rankings). In combinations, order does not matter (e.g., card hands, team selections). Permutations give larger numbers than combinations for the same n and r.

The permutation formula is P(n,r) = n! / (n-r)!. It calculates the number of ways to arrange r items from n total items where order matters. For example, P(10,3) = 720 ways to arrange 3 items from 10.

The combination formula is C(n,r) = n! / (r! × (n-r)!). It calculates the number of ways to select r items from n total items where order does not matter. For example, C(10,3) = 120 ways to choose 3 from 10.

Permutation with repetition is n^r — the number of ways to arrange r items from n when each item can be reused. For example, a 3-digit lock with 10 digits has 10³ = 1,000 possible combinations.

Combination with repetition is C(n+r-1, r) = (n+r-1)! / (r! × (n-1)!). It counts ways to select r items from n types when repetition is allowed, like choosing candies from a jar.

Factorials grow exponentially; 171! exceeds the maximum double-precision floating point number capacity (approximately 1.8 × 10³⁰⁸) in computer architectures. Beyond 170, the results cannot be represented accurately.

Use permutations when the arrangement order matters (passwords, race results, seating). Use combinations when only the selection matters (team picks, lottery numbers, committee formation). If in doubt, ask: "Does swapping two items create a different outcome?"

C(n,k) is also written as "n choose k" or (n k) — the binomial coefficient. It represents the number of ways to choose k items from n without regard to order. It appears in Pascal's triangle and the binomial theorem.

The factorial n! = n × (n-1) × (n-2) × ... × 2 × 1. For example, 5! = 120. By convention, 0! = 1. Factorials grow extremely fast — 10! = 3,628,800.

Applications include lottery probability calculations, password security analysis, genetic combinations, tournament scheduling, quality control sampling, and probability theory in statistics.

Yes. For permutations of a multiset with n total items where n₁, n₂, ..., nₖ are identical items, the formula is n! / (n₁! × n₂! × ... × nₖ!). This accounts for indistinguishable arrangements.

Each entry in Pascal's triangle is a combination number: C(n,k). Row n contains C(n,0), C(n,1), ..., C(n,n). The recursive property C(n,k) = C(n-1,k-1) + C(n-1,k) generates the triangle.

Yes, this free permutation combination calculator is 100% free with no signup required. Calculate nPr, nCr, and repetition variants online with instant step-by-step results.

The Permutation & Combination Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Permutation & Combination Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.