Permutation & Combination Calculator

Find the number of ways to arrange (nPr) or select (nCr) items from a set with step-by-step factorial working.

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Enter the total number of items in the set (n) and the number of items to choose (r) to calculate the arrangements and selections.

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nCr

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A Guide to Permutations and Combinations

In the branch of mathematics known as combinatorics, we often need to find the number of ways to select or arrange items from a larger set. To choose the correct method, you must answer one simple question: "Does the order of the items matter?" This distinction separates permutations (where order is critical) from combinations (where order is completely ignored).

Arrangements vs. Selections

A permutation (nPr) counts the number of unique arrangements. For example, if you are picking a President, Vice President, and Secretary, the order matters because each position is different. A combination (nCr) counts the number of unique groups. If you are picking a three-person committee, the order does not matter because they all share the same role.

Knowing the number of possible outcomes is the first step before calculating likelihoods, which you can resolve using our calculating odds and chances tool. To pick random values from your groups, check out our generating random values tool. For broader database trends, view our comprehensive statistics calculators tool.

Everyday Matching Scenarios

Lock Combinations: Standard padlock codes are actually permutations, because entering the correct digits in the wrong order will not open the lock.
Lottery Ticket Picks: Lottery drawings are combinations, because you win if you match the numbers regardless of the order they were drawn. You can check rates with our percentage calculators tool.
Race Finishes: The number of ways runners can finish in Gold, Silver, and Bronze positions is a permutation task. You can do quick divisions with our simple daily math tool.
Schedule Rotations: Planning how to assign shifts to employees involves calculating combinations to build unique worker teams.

The Power of Factorials

Both formulas rely on factorials (represented by the exclamation mark symbol !). A factorial means multiplying a positive integer by every whole number below it down to one (for example, 4! is 4 × 3 × 2 × 1 = 24).

Because factorials grow extremely fast, calculating nPr or nCr manually for sets larger than ten can become tedious. Our calculator handles large values instantly, showing you the exact calculations and factorials.

It is also useful to consider if repetition is allowed in your arrangements. For example, if you are creating a password where characters can be reused, the number of possibilities is calculated as n raised to the power of r, which produces far more combinations than a standard permutation where each item can only be chosen once.

Example of Picking a Committee

Suppose a club has 5 members, and they want to select a committee of 3 people.

Since the order does not matter, we use the combination formula (5C3). The formula is: 5! / (3! × (5 - 3)!) = 120 / (6 × 2) = 120 / 12 = 10. There are exactly 10 unique ways to form the committee. If the roles were ordered (like President, Treasurer, Secretary), we would use the permutation formula (5P3), yielding 5! / (5 - 3)! = 120 / 2 = 60 arrangements. This shows how order alters the mathematical outcomes.