Multinomial Coefficient Calculator

Enter n (total count) and your k values (comma-separated group sizes like 2,3,5) to calculate the multinomial coefficient — the number of ways to partition n objects into ordered groups. The calculator returns the coefficient value along with the full factorial breakdown, so you can verify every step of the computation. Also try the Possible Combinations Calculator.

The total number of objects or the exponent. Must equal the sum of all k values.

Enter the group sizes separated by commas. They must sum to n. Works as a binomial coefficient if only one k value is entered.

Results

Multinomial Coefficient

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n (Sum of k values)

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Number of Groups (j)

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n! (Numerator)

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k1! × k2! × … × kj! (Denominator)

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Results Table

Frequently Asked Questions

What is a multinomial coefficient?

A multinomial coefficient counts the number of ways to partition n objects into j ordered groups of specified sizes k1, k2, …, kj. It is calculated as n! divided by the product of each group's factorial: n! / (k1! × k2! × … × kj!). The sum of all k values must equal n. See also our Permutation without Repetition Calculator.

What is the multinomial coefficient formula?

The formula is: C(n; k1, k2, …, kj) = n! / (k1! × k2! × … × kj!), where n = k1 + k2 + … + kj. For example, the multinomial coefficient for n=10 with groups 2, 3, 5 is 10! / (2! × 3! × 5!) = 2520.

How is the multinomial coefficient different from the binomial coefficient?

The binomial coefficient C(n, k) is a special case of the multinomial coefficient with exactly two groups: k and n−k. The multinomial coefficient generalizes this to any number of groups j, making it applicable to polynomial expansions involving more than two terms.

What is the multinomial coefficient used for?

Multinomial coefficients appear in combinatorics, probability, and algebra. They are used to expand multinomial expressions like (a + b + c)^n, to compute probabilities in the multinomial distribution, and to count arrangements of objects split into distinct categories. You might also find our find Number of Combinations C(n,r) with Combination without Repetition Calculator useful.

How do I use this calculator?

Enter the total n in the first field and your comma-separated k values in the second field (e.g. 2,3,5 for n=10). The calculator will verify that the k values sum to n, then compute the coefficient, the factorial breakdown, and a per-group table automatically.

Can this calculator be used as a binomial coefficient calculator?

Yes. If you enter only two k values that sum to n — for example n=6 and k values 2,4 — the result equals the standard binomial coefficient C(6,2) = 15. Entering a single k value equal to n gives a coefficient of 1.

What happens if my k values don't sum to n?

The calculator will display an error message indicating that the k values must sum to n. You need to either adjust the k values or update n to match the sum before a valid result can be computed.

Is there a Pascal's Triangle equivalent for multinomial coefficients?

For binomial coefficients, Pascal's Triangle provides a simple recursive pattern. For multinomial coefficients, there is an analogous Pascal's Simplex (or Pascal's Pyramid for trinomials), where each entry equals the sum of adjacent entries from the layer above, but it becomes increasingly complex as the number of terms grows.