Math July 13, 2026 · 8 Min Read

Permutation & Combination Calculator – Guide & Formulas

Calculate standard permutations and combinations with or without repetitions using step-by-step factorial formulas.

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Put these formulas into practice with our instant, step-by-step Permutation & Combination Calculator.

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The Permutation & Combination Calculator resolves core combinatorics problems instantly. It computes standard variations (nPr, nCr) and repetitive models with rigorous mathematical steps. Whether you need to calculate permutations, find combinations, or determine how many ways to arrange or select items, this free online tool provides accurate results with step-by-step factorial breakdowns.

Key Takeaway

Use the free Permutation & Combination Calculator to calculate standard permutations and combinations with or without repetitions using step-by-step factorial formulas. Get instant results with step-by-step explanations.

How to Use the Permutation & Combination Calculator

  1. Enter the total items count (n) up to 170.
  2. Enter the selected items count (r) up to 170.
  3. Ensure inputs are non-negative integers with r ≤ n.
  4. Review C(n,r) and P(n,r) values alongside repetition equivalents.
  5. Copy or share your permutation and combination results.

The Formula

P(n, r) = n! / (n-r)! ; C(n, r) = n! / (r! * (n-r)!)

Variable Definitions

  • n!: Factorial representing product of all positive integers up to n
  • r: The subset of elements chosen from the collection
  • P(n,r): Permutations — ordered arrangements of r items from n
  • C(n,r): Combinations — unordered selections of r items from n

Selecting 3 items from a pool of 10

Calculate permutations and combinations for n = 10 and r = 3.

  1. Step 1: P(10, 3) = 10! / (10-3)! = 10! / 7! = 10 * 9 * 8 = 720.
  2. Step 2: C(10, 3) = P(10, 3) / 3! = 720 / 6 = 120.
  3. Step 3: Permutations with repetition n^r = 10^3 = 1,000.
  4. Step 4: Combinations with repetition (10+3-1)! / (3! * 9!) = 12! / (6 * 9!) = 220.

Financial Advisory Notice

This calculator is for educational and informational purposes only. Always verify critical calculations with authoritative sources.

Frequently Asked Questions

How do I use the permutation and 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.

What is the difference between permutations and combinations?

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.

What is the permutation formula?

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.

What is the combination formula?

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.

What is permutation with repetition?

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.

What is combination with repetition?

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.

Why is n limited to 170?

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.

When should I use permutations vs combinations?

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?"

What is n choose k notation?

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.

How does the factorial function work?

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.

What are real-world applications of permutations and combinations?

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

Can I calculate permutations of a multiset?

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.

What is the relationship between combinations and Pascal's triangle?

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.

Is this permutation combination calculator free?

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.