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Permutation and Combination Calculator

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Enter n (total items) and r (items chosen) to find the number of permutations and combinations.

Permutation and Combination Calculator

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Permutations (nPr) order matters
720
Combinations (nCr) order doesn't matter
120

The formulas

Permutations: nPr = n! / (n - r)! Combinations: nCr = n! / (r! × (n - r)!)
Example

Choosing 3 items from 10: permutations (order matters) = 10!/(10-3)! = 720; combinations (order doesn't matter) = 720/3! = 120.

Step-by-step guide

  1. Enter n, the total number of items available.
  2. Enter r, the number of items being chosen.
  3. Read both results - permutations if order matters, combinations if it doesn't.

Why permutations are always larger than combinations

Permutations count every possible ordering as a distinct outcome (choosing A-then-B is different from B-then-A), while combinations treat those as the same single outcome. Since combinations divide out exactly the number of ways to order each group of r items (r! ways), permutations are always larger than or equal to combinations for the same n and r - and the gap between them grows quickly as r increases, since the number of ways to reorder a larger group grows factorially.

Common mistakes

Using permutations when order genuinely doesn't matter (like choosing a committee) or combinations when it does (like ranking finishers in a race) - picking the wrong one gives a very different answer.
Entering r greater than n - you can't choose more items than exist in the total group, so this input combination is invalid.

Frequently asked questions

How do I know whether to use permutations or combinations?

Ask whether order matters for your situation - picking a president, VP, and treasurer from a group (each role is distinct) calls for permutations; picking 3 people for a committee with no distinct roles calls for combinations.

What if r equals n?

Permutations become n! (every possible full ordering of all items), while combinations become exactly 1 (there's only one way to "choose" the entire group, regardless of order).

What if r is 0?

Both permutations and combinations equal 1 - there's exactly one way to choose nothing from any group, by definition.

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