Counting, Permutations & Combinations

Multinomial Coefficient Calculator

Count assignments of distinct objects to labeled groups of specified sizes.

Counting & Combinatorics

Enter the labeled group sizes

Exact multinomial coefficient

Try an example

Group sizesEnter one non-negative group size per line.

Exact coefficient

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Understand the subject

A multinomial coefficient counts assignments of distinct objects to labelled groups of fixed sizes. Order inside each group does not matter.

Use it for splitting people or objects into named teams, categories, or bins.

The relationship

n! ÷ (k₁! k₂! …)

See the structure

choose 3then choose 2last 1 remains6! ÷ (3!2!1!) = 60

Worked example

Split 6 objects into labelled groups of 3, 2, and 1: 6! ÷ (3!2!1!) = 60.

Interpret with care

If group labels do not matter, this is not the correct model.

Use the Permutation & Combination Generator when a small case needs to be inspected rather than only counted.

Quick guide

Calculation method

For group sizes k₁ through kₘ totaling n, divide n! by the product k₁!⋯kₘ!.

Worked example

The group labels matter, while order inside each group does not.

6!/(3!2!1!) = 60

Supported inputs

Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.

Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.