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Multinomial Coefficient from Three Counts Calculator

(k₁ + k₂ + k₃)! / (k₁! · k₂! · k₃!).

Compute the multinomial coefficient C(n; k₁, k₂, k₃) = n! / (k₁! · k₂! · k₃!), where n = k₁ + k₂ + k₃. This counts the distinct ways to partition n distinguishable objects into three labelled groups of sizes k₁, k₂, k₃; equivalently the number of distinct permutations of a multiset with three element types.

Daniel Oh By Daniel Oh
Published Last reviewed 1 min read

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  1. Fill in the inputs above using the units you already have.
  2. Values update automatically as you type — no submit button needed.
  3. Hover any result row for the underlying formula and intermediate values.

Formula

(k₁ + k₂ + k₃)! / (k₁! · k₂! · k₃!).

In depth

Compute the multinomial coefficient C(n; k₁, k₂, k₃) = n! / (k₁! · k₂! · k₃!), where n = k₁ + k₂ + k₃. This counts the distinct ways to partition n distinguishable objects into three labelled groups of sizes k₁, k₂, k₃; equivalently the number of distinct permutations of a multiset with three element types.