CALCULATE WITH CONFIDENCE

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
Enter one non-negative group size per line.

Exact coefficient

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Enter valid values to see the result.

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Quick guide

How to use this calculator

  1. Enter one non-negative size for each labeled group.
  2. The calculator derives the total number of objects.
  3. Read the exact multinomial coefficient.

Calculation method

Choose each labeled group in sequence

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

Worked example

Split six objects into groups of 3, 2, and 1

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

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

Supported inputs

Precision and limits

Discrete domain

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

Exact integers

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