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Multiset Permutation Calculator

Use this page when you are arranging all items, but some of those items are identical to each other. You can either enter item-group counts directly, or type a word, number string, or emoji string into the optional entry box and let the calculator derive the counts.

Result

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What your result means

The explanation updates with the current inputs.

Calculation work

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    Previewing a small sample keeps the page fast and readable.

    Calculators Order matters, duplicates collapse

    Count unique orderings when some items are identical copies, so swapping matching copies does not create a new arrangement.

    Practice this formula See exact answer pages

    When to use this

    Use this formula when the question matches this rule set

    Use multiset permutation when all positions are filled, order matters, and at least one item type repeats. Typical cases are repeated letters (like APPLE, BANANA, BALLOON), duplicate tiles, or repeated symbols. Do not use this when you are only choosing a subset; that is combination/permutation n and r logic instead.

    What this result means

    Interpret the output, not just the number

    The result is the number of different final arrangements you can actually see. Arrangements that differ only by swapping identical copies are counted once, not multiple times.

    Formula: T! / (n1! n2! n3! ...)

    Calculator inputs

    Know what to enter in each field

    This page uses one field because the counts already capture everything needed: how many copies exist of each distinct type before arranging.

    Item-group counts

    Use this when you are arranging all items and some items are identical copies before arranging.

    What to enter: Enter one whole number per distinct item type, separated by commas.

    How to use it: Step 1: Count each distinct type. Step 2: Write how many copies each type has. Step 3: Enter those counts (including 1s for singletons). The order of the counts does not matter.

    Example: BALLOON -> 2,2,1,1,1 because L repeats twice, O repeats twice, and B, A, N are singles.

    Optional item to count entry

    Type a source string and let the calculator derive item-group counts automatically.

    What to enter: Enter a word, number string, or emoji string.

    How to use it: The calculator counts repeated graphemes and auto-populates the item-group counts field. Spaces are ignored.

    Example: APPLE -> 2,1,1,1, 112233 -> 2,2,2, 😀😀😎😎😎🔥 -> 3,2,1.

    Item examples

    A quick dropdown that seeds the optional entry field.

    What to enter: Choose any sample value from letters, numbers, or emoji strings.

    How to use it: Selecting an example writes it into the optional entry box, then auto-fills item-group counts and recalculates.

    Example: Choose BALLOON, 9000099, or 😀😀😎😎😎🔥 to see each family.

    Worked examples

    Quick checks with realistic inputs

    APPLE arrangements

    Count each distinct letter first: P appears 2 times, while A, L, and E appear once. Enter 2,1,1,1.

    For item-group counts 2, 1, 1, 1, the number of distinct arrangements is 60.

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    112233 arrangements

    Number string example: 112233 has three types repeated twice each, so the counts are 2,2,2.

    For item-group counts 2, 2, 2, the number of distinct arrangements is 90.

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    😀😀😎😎😎🔥 arrangements

    Emoji string example: 😎 appears 3 times, 😀 appears 2 times, and 🔥 appears once, so the counts are 3,2,1.

    For item-group counts 3, 2, 1, the number of distinct arrangements is 60.

    Load this example into the calculator