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MasterMath

Permutation Calculator

How many ways a set of items can be ordered, allowing for repeated items — which cut the total sharply — and for arrangements in a circle.

Distinct orderings

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Distinct orderings—
If all were different—
Divided by—
In a circle—

How this was worked out

    The rule

    n! ÷ (r₁! × r₂! × …) for repeated items

    Why it works

    With all items distinct there are n! orderings. Repeated items mean some of those orderings are indistinguishable, so you divide by the factorial of each repeat count to remove the duplicates.

    How to do it by hand

    1. Work out the factorial of the total number of items
    2. For each repeated item, work out the factorial of how many times it repeats
    3. Divide the first by the product of the rest
    4. For a circle, divide by n, since rotations are the same arrangement

    What is worth knowing

    The word MISSISSIPPI is the standard example: eleven letters, but with four S, four I and two P, so 11! divided by 4!4!2! gives 34,650 rather than 39,916,800 — over a thousand times fewer. Circular permutations divide by n for a different reason: seating five people round a table has 4! distinct arrangements, not 5!, because rotating everyone one seat leaves the arrangement unchanged. If the table can also be flipped, you halve it again.

    Frequently asked questions

    What is a permutation?

    An arrangement where order matters. n distinct items have n! permutations.

    Why divide by the repeats?

    Because swapping two identical items gives the same arrangement. Dividing removes the duplicates.

    Why are circular permutations fewer?

    Because rotating everyone by one place gives the same arrangement. Five people round a table have 4! arrangements, not 5!.

    What is the MISSISSIPPI example?

    Eleven letters with four S, four I and two P: 11! ÷ (4!4!2!) = 34,650 distinct arrangements.