Multinomial Coefficient Calculator
Count the arrangements of a multiset with the multinomial coefficient n! / (k₁! k₂! …) — e.g. the distinct orderings of the letters in MISSISSIPPI.
How it works
C(n; k₁, k₂, …, kₘ) = n! / (k₁! · k₂! · … · kₘ!), n = Σ kᵢ
The multinomial coefficient counts the distinct orderings of n items when items within each group are identical — it generalises “n choose k” from two groups to any number. Dividing n! by the factorial of each group size removes the arrangements that only shuffle identical items. This is the go-to formula for anagrams of a word with repeated letters and for splitting n objects into labelled boxes of fixed sizes.
Worked example
MISSISSIPPI has 11 letters with counts M1, I4, S4, P2. Arrangements = 11! / (1! 4! 4! 2!) = 39,916,800 / 1,152 = 34,650.
Sources: Multinomial theorem — Wikipedia
Frequently asked questions
How does this relate to the binomial coefficient?
With just two groups, C(n; k, n−k) is exactly the binomial coefficient C(n, k). The multinomial simply extends it to three or more groups.
What do the group sizes represent?
They are the counts of identical items in each category — for a word, the number of times each distinct letter repeats; for boxes, how many items go in each box. They must add up to n.
Why divide by each group’s factorial?
Rearranging identical items within a group produces no new distinct arrangement. Dividing by kᵢ! for each group cancels those indistinguishable orderings.
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This calculator is for educational purposes. Double-check important results before acting on them.