Compute P(n,r), n!, anagrams, circular permutations and derangements, with the working and a list of every case.
A permutation counts the ordered arrangements of the elements of a set, where order matters: AB and BA are different outcomes. Choosing r out of n elements without repeating gives P(n, r) = n! / (n − r)!, also called a partial permutation or arrangement; ordering all n gives n!. With repetition each of the r positions has n options and the total is nʳ. The calculator uses BigInt and descending products (no giant factorials), accepts n up to 5000 and also covers anagrams of words with repeated letters (n! divided by the factorial of each repeat), circular permutations (n−1)!, derangements !n and the combination versions. It shows the formula, the step by step working, the five results for the same n and r, a list of every case in order (exportable as TXT and CSV) and turns the total into a probability, with password, PIN and anagram examples.
Without repetition, P(n, r) = n! / (n − r)! = n · (n−1) · … · (n − r + 1). In practice it is a product of r descending factors, and the tool computes exactly that without building the whole n!. If r = n, P(n, n) = n!. The link to combinations is P(n, r) = C(n, r) · r!.
With repetition, each position picks among n options independently: PR(n, r) = nʳ. When some items are identical (anagrams), divide n! by the factorial of each repeated count: n! / (n₁! · n₂! · … · nₖ!). For circular permutations fix one element and order the other n − 1: (n−1)!.
A podium with 8 runners (3 ordered places): P(8, 3) = 8 · 7 · 6 = 336. Anagrams of ANANAS: 6 letters with A three times and N twice, so 6! / (3! · 2! · 1!) = 720 / 12 = 60. Anagrams of MISSISSIPPI: 11! / (4! · 4! · 2! · 1!) = 34,650.
A round table with 5 people: (5 − 1)! = 24 seatings. Secret Santa with 5 people where nobody draws themselves: !5 = 44 valid draws out of 120 possible (about 36.7%). As n grows that share approaches 1/e, about 36.79%.
Without repetition r cannot exceed n. With repetition it can, and nʳ grows quickly: an 8-character password over 62 symbols has 62⁸ = 218,340,105,584,896 possibilities. The tool shows the scientific notation and digit count for huge results and copies the whole number.
Permutations show up in sorting algorithm analysis, cryptography and password counting, task scheduling, card games and any problem where the final order changes the answer.
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src="https://jkit.tools/embed/en-US/permutation-calculator"
width="100%"
height="600"
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loading="lazy"
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></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
A permutation is an ordered arrangement of elements, where order matters. When you order all n elements, the total is n! (5 different books on a shelf: 5! = 120). When you choose and order only r of them, it is a partial permutation, P(n, r) = n! / (n − r)!. Example: 3 ordered letters out of A, B, C and D gives P(4, 3) = 4 · 3 · 2 = 24.
Choose r of n, ordered. E.g. podium, queue.
P(n, r) = n! / (n − r)!
720
Multiplies the total. Use 1 (Powerball: 26).
Click a cell to use n and k.
All calculations stay in your browser. No data is sent to any server.