Permutation & Combination Calculator

Calculate permutations (nPr) and combinations (nCr) instantly for any n and r. Free online calculator with exact factorial-based results and step-by-step solutions, no precision loss for large numbers.

What is Permutation & Combination Calculator?

Permutations and combinations are the two fundamental ways of counting how many ways you can select or arrange items from a larger set — a cornerstone of probability, statistics, and combinatorics.

A permutation, written P(n,r) or nPr, counts the number of ways to arrange r items chosen from n, where the order of selection matters. A combination, written C(n,r) or nCr, counts the number of ways to choose r items from n, where order doesn't matter. This calculator computes both using exact arbitrary-precision arithmetic, so results stay perfectly accurate even for large factorials.

Formula

Permutation: P(n,r) = n! / (n-r)!

Combination: C(n,r) = n! / (r!(n-r)!)

Where n! (n factorial) = n × (n-1) × (n-2) × ... × 2 × 1

How to Calculate

Enter n, the total number of items in the set.
Enter r, the number of items you're selecting.
The calculator computes both P(n,r) and C(n,r) using the factorial formulas.
Review the full step-by-step breakdown of each formula.

Example

For n=10, r=3: P(10,3) = 10!/(10-3)! = 10 × 9 × 8 = 720 ways to arrange 3 items from 10 in order. C(10,3) = 10!/(3!×7!) = 720/6 = 120 ways to choose 3 items from 10 without regard to order.

Key Benefits

Computes both permutations and combinations in one place
Exact results using arbitrary-precision arithmetic — no rounding errors on large numbers
Full step-by-step breakdown of the factorial formulas
Clear explanation of when to use each formula

Common Mistakes to Avoid

Using the permutation formula when order doesn't actually matter in the problem (or vice versa)
Forgetting that r cannot exceed n
Miscounting when a problem allows repetition — these formulas assume items are chosen without repetition
Confusing nPr and nCr notation when reading a textbook or exam question

Pro Tips

Ask yourself "does the order matter?" first — that single question determines whether to use permutations or combinations
For large n and small r, it's faster to compute C(n,r) as n×(n-1)×...×(n-r+1) / r! rather than using full factorials
Committee, team, or group selection problems are almost always combinations
Ranking, arranging, or ordering problems (like race results or seating) are almost always permutations

Key Terms Explained

Factorial (n!): The product of all positive integers up to n
Permutation: An ordered arrangement of a subset of items
Combination: An unordered selection of a subset of items
Binomial Coefficient: Another name for C(n,r), written as (n choose r)

Common Use Cases

Calculating lottery or raffle odds
Solving probability and combinatorics homework
Counting possible passwords, PINs, or arrangements
Determining the number of possible committees, teams, or groupings

Frequently Asked Questions

What's the difference between a permutation and a combination?

In a permutation, order matters — arranging A, B, C differently counts as different outcomes. In a combination, order doesn't matter — choosing the same set of items in any order counts as one outcome. Permutations always give a result greater than or equal to combinations for the same n and r.

What does P(n,r) mean?

P(n,r), also written nPr, is the number of ways to arrange r items selected from a set of n items, where the order of selection matters. It's calculated as n! / (n-r)!.

What does C(n,r) mean?

C(n,r), also written nCr or "n choose r," is the number of ways to select r items from a set of n items where order doesn't matter. It's calculated as n! / (r!(n-r)!).

Why is 0! equal to 1?

By mathematical convention, 0! = 1. This isn't arbitrary — it makes the permutation and combination formulas produce correct results (like C(n,0) = 1, meaning there's exactly one way to choose nothing) and is consistent with the recursive definition n! = n × (n-1)!.

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