How do I know whether to use permutation or combination?

Ask whether changing the order creates a new outcome. If yes, use a permutation. If no, use a combination. First, second, and third place is a permutation because ABC and BAC are different results. A three-person study group is a combination because ABC and BAC contain the same people.

Why is nPr usually larger than nCr?

A permutation counts every ordering separately, while a combination groups those orderings together. For example, choosing 3 people from 10 has fewer combinations than ordered arrangements because each group of 3 can be arranged in 3! different ways.

The exclamation mark means factorial. For a positive whole number n, n! means n × (n−1) × (n−2) × ... × 1. For example, 5! = 120. Factorials grow very quickly, which is why counting problems can become large even with moderate n.

Yes. If r = n, then nPr = n! because you are arranging all n items. For combinations, nCr = 1 because there is only one way to choose all items when order is ignored.

Does this calculator handle repetition?

This calculator handles the common classroom case without repetition. That means each item can be chosen only once. Some advanced problems allow repetition, such as PIN codes where digits can repeat; those use different formulas.

Education

Permutation and Combination Difference Calculator

Compare permutations and combinations for the same n and r values. The calculator shows nPr and nCr side by side so students can see how “order matters” changes the answer.

Permutation vs combination: the question is order

Students usually struggle with permutations and combinations because the arithmetic looks similar, but the story is different. A permutation counts arrangements where order matters. A combination counts groups where order does not matter. OpenStax explains this distinction in its exact section on counting principles, permutations, and combinations.

If you are choosing president, vice president, and secretary, order matters because each role is different. If you are choosing a 3-person committee with no roles, order does not matter. This calculator shows both values side by side so the difference becomes obvious.

Formulas

Permutation: nPr = n! ÷ (n − r)! Combination: nCr = n! ÷ [r!(n − r)!]

The combination formula divides by r! because the same group can be arranged in r! different orders. When the order does not matter, those repeated arrangements need to be removed.

Quick decision table

Situation	Use
Race finish order	Permutation
Committee selection	Combination
Password arrangements	Permutation-style thinking
Lottery numbers where order is ignored	Combination

Frequently asked questions

Ask whether changing the order creates a new outcome. If yes, use a permutation. If no, use a combination. First, second, and third place is a permutation because ABC and BAC are different results. A three-person study group is a combination because ABC and BAC contain the same people.
A permutation counts every ordering separately, while a combination groups those orderings together. For example, choosing 3 people from 10 has fewer combinations than ordered arrangements because each group of 3 can be arranged in 3! different ways.
The exclamation mark means factorial. For a positive whole number n, n! means n × (n−1) × (n−2) × ... × 1. For example, 5! = 120. Factorials grow very quickly, which is why counting problems can become large even with moderate n.
Yes. If r = n, then nPr = n! because you are arranging all n items. For combinations, nCr = 1 because there is only one way to choose all items when order is ignored.
This calculator handles the common classroom case without repetition. That means each item can be chosen only once. Some advanced problems allow repetition, such as PIN codes where digits can repeat; those use different formulas.