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Permutation and Combination Calculator

Compute permutations (nPr) and combinations (nCr) with exact big-number results. Includes factorials, with/without repetition options, and step-by-step formulas you can copy into homework or reports.

nPr / nCr BigInt exact With repetition Steps

nPr & nCr Calculator

Enter n and r, choose the counting type, and get exact results instantly.

Outputs are exact integers using BigInt. If you choose very large n or r, results can be extremely long.
Factorials grow very fast. Big inputs can produce extremely long results.

Order matters vs order does not matter

Use permutations (nPr) when the arrangement is important (ABC is different from BAC). Use combinations (nCr) when you only care which items are chosen (ABC is the same selection as BAC).

Scenario Order matters? Use Formula
Pick 3 winners from 10 in ranked order (1st/2nd/3rd) Yes Permutation nPr = n! / (n−r)!
Choose 3 people from 10 for a team No Combination nCr = n! / (r!(n−r)!)
PIN code of length r from n digits (digits can repeat) Yes Permutation with repetition n^r
Choose r items from n types with repeats allowed No Combination with repetition (n+r−1)Cr
“With repetition” means you can reuse items. “Without repetition” means once you pick an item, it cannot be picked again.

Permutation and Combination Calculator: When to Use nPr vs nCr

A permutation and combination calculator helps you solve counting problems without doing large factorial math by hand. The two core ideas are simple: permutations count arrangements (order matters) and combinations count selections (order does not matter). These appear everywhere: probability, statistics, password counting, seating plans, lottery rules, and sample-size calculations.

Permutations (nPr): Order Matters

A permutation answers: “How many ways can I choose r items from n and arrange them?” If you are ranking winners, arranging books on a shelf, or assigning unique roles, you are in permutation territory.

nPr = n! / (n − r)!

The factorial (n!) means multiply every whole number from 1 to n. Because factorials grow extremely fast, this tool uses BigInt to keep results exact even when the numbers become very large.

Permutations with repetition

If repetition is allowed (you can reuse items), the count changes. For sequences of length r where each position has n choices:

Permutations with repetition = n^r

Combinations (nCr): Order Does Not Matter

A combination answers: “How many ways can I choose r items from n where order does not matter?” If you are selecting a committee or choosing lottery numbers, you are in combination territory.

nCr = n! / (r!(n − r)!)

Combinations with repetition

When repeats are allowed and order does not matter (like selecting scoops of ice cream flavors where a flavor can repeat), use combinations with repetition:

Combinations with repetition = (n + r − 1)Cr

Why nPr and nCr Matter in Probability

Many probability questions depend on counting how many outcomes exist. For example, drawing hands from a deck, selecting groups from a population, or counting arrangements of events can all require nCr or nPr. Once you can count outcomes correctly, probability becomes:

Probability = (favorable outcomes) / (total outcomes)

Step-by-step Strategy to Pick the Right Formula

Quick decision checklist

  1. Identify n: how many total items/types exist.
  2. Identify r: how many you are choosing/placing.
  3. Ask: does order matter? If yes → permutations. If no → combinations.
  4. Ask: can items repeat? If yes → use “with repetition” formulas.

Common Mistakes

  • Mixing up order: “ABC” vs “BAC” changes the count only when order matters.
  • Forgetting repetition rules: If you cannot reuse items, use the “without repetition” setting.
  • Using factorials directly for large values: Factorials explode in size. BigInt keeps results exact, but outputs can be huge.

FAQ

Permutation and Combination Calculator FAQs

Answers to common questions about nPr, nCr, repetition, and factorials.

Permutations count arrangements where order matters. Combinations count selections where order does not matter.

nPr is the number of ways to choose and arrange r items from n. Formula: nPr = n! / (n−r)!.

nCr is the number of ways to choose r items from n when order does not matter. Formula: nCr = n! / (r!(n−r)!).

Yes. It uses BigInt arithmetic for exact integer results. For very large inputs, outputs can be extremely long.

With repetition means you can select the same item multiple times. For combinations with repetition: (n+r−1)Cr. For permutations with repetition of length r: n^r.

Without repetition, nPr and nCr are 0 when r > n. With repetition, combinations/permutations are still defined.

Factorials grow very quickly. This tool uses BigInt so it can compute exact results far beyond normal floating-point limits.

Use combinations for selection problems like choosing a team, picking toppings, or selecting lottery numbers where order does not matter.

Use permutations for arrangement problems like seating order, PIN codes, rankings, or any case where order matters.

Results are mathematical counts based on the chosen rules (order, repetition). Always match the calculator settings to the wording of the problem.

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