Permutation Calculator
nPr — arrangements, where order changes the answer.
Work out Permutation. nPr — arrangements, where order changes the answer. Free, with no account and nothing to download.
Ways to arrange 3 of 10
720
120 combinations × 6 orderings each
Permutations count arrangements, so ABC and CBA are different outcomes. That is the right model for finishing positions, passwords and seating, and the wrong one for lottery tickets and committees, where order is irrelevant. nPr = n!/(n−r)!, and it is always r! times the corresponding combination — each selection can be ordered r! ways. The distinction is the whole of the topic: deciding whether order matters is harder than the arithmetic, and it is where nearly every wrong answer comes from.
How the Permutation Calculator works
Enter a total and a number of positions for the count of ordered arrangements. Deciding whether order matters is harder than the arithmetic, and it is where nearly every wrong answer in this topic comes from — so the page shows both figures side by side.
Also known as: npr calculator · number of arrangements calculator · ordered selection calculator · permutations formula
Frequently asked questions
When should I use permutations rather than combinations?
When rearranging the same items gives a different outcome. Finishing positions, passwords, seating plans and lock codes are permutations. Committees, lottery tickets and card hands are combinations.
How do I calculate permutations?
nPr = n!/(n−r)!, which is n × (n−1) × … for r factors. It is always r! times the corresponding combination, because each selection can be ordered r! different ways.
Why is a combination lock really a permutation lock?
Because the order genuinely matters — 1-2-3 does not open a lock set to 3-2-1. The everyday name is mathematically backwards, which is a useful mnemonic for the distinction.
What if items can repeat?
Then it is n^r rather than nPr, since every position has all n choices available. A four-digit PIN has 10⁴ = 10,000 possibilities, not 10P4 = 5,040.
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