Permutation and Combination Calculator
Calculate ordered permutations and unordered combinations for n items taken r at a time
Combinations nCr
How we calculated it
About the Permutation and Combination Calculator
Permutations and combinations both count ways of choosing items, but they answer different questions. Permutations care about order; combinations do not.
Choosing the wrong one can multiply the answer dramatically, so the first step is deciding whether two selections with the same items in a different order should count as different outcomes.
How to Use the Permutation and Combination Calculator
- Enter the total number of available items n.
- Enter the number chosen r.
- Choose permutation when order matters or combination when order does not.
- Calculate and check that 0≤r≤n for ordinary selection problems.
How the Calculation Works
Permutations use P(n,r)=n!÷(n−r)!. Combinations use C(n,r)=n!÷[r!(n−r)!].
The additional r! in the combination denominator removes the different internal orderings of the same selected group.
Worked example
Permutation and Combination Calculator Worked Example
Choose 3 from 5 items
Permutations: 5P3 = 5×4×3 = 60 ordered arrangements. Combinations: 5C3 = 60÷3! = 60÷6 = 10 unordered groups.
Result guide
Understanding Your Permutation and Combination Calculator Result
If rearranging the chosen items creates a new outcome, use permutations. If the group is considered the same regardless of order, use combinations. In probability problems, the distinction affects both the numerator and denominator of an outcome count, so choosing the wrong model can change the final probability by a large factor.
Before you rely on the answer
Assumptions and Limitations
- Items are distinct unless the problem states otherwise.
- Selections are made without replacement under the standard formulas.
- n and r are non-negative integers with r≤n.
Practical tip
Using This Result in Practice
Translate the problem into plain language before choosing a formula. “Choose a committee” usually suggests combinations, while “assign first, second and third place” suggests permutations. Ask whether ABC and CBA should count as the same selection. If yes, you are usually looking at a combination rather than a permutation.
Related calculations
Useful Calculations to Continue With
Both formulas depend on factorials, so use the Factorial Calculator if you want to inspect the underlying products.
If these counts are being used to build a probability model, use the Probability Calculator after identifying the number of outcomes.
Common questions
Permutation and Combination Calculator FAQ
When should I use combinations?
Use combinations when order does not matter, such as choosing a committee from a group.
When should I use permutations?
Use permutations when different orders count as different outcomes, such as arranging people in ranked positions.
Why is nCr smaller than nPr?
Combinations collapse the r! different orderings of each selected group into one outcome.