Multinomial Coefficient Calculator
The multinomial coefficient counts how many distinct ways n objects can be split into ordered groups of given sizes. Enter up to four group sizes and the calculator sums them to find n, then returns n! divided by the product of the group factorials. This is the count of distinguishable arrangements of a multiset, the coefficient in a multinomial expansion, and the combinatorial core of the multinomial probability distribution. It generalises the familiar binomial "n choose k" to any number of categories, which makes it indispensable in combinatorics, algebra, and probability.
Multinomial coefficient formula
n = k1 + k2 + k3 + k4
Coefficient = n! / (k1! * k2! * k3! * k4!)
Each group size must be a non-negative whole number. The result is the number of distinct ways to assign n labelled items so that k1 go to the first category, k2 to the second, and so on. Unused groups are set to 0, whose factorial is 1.
Worked examples and uses
- The word MISSISSIPPI (1 M, 4 I, 4 S, 2 P) has 11! / (1! 4! 4! 2!) = 34,650 distinct arrangements.
- With two groups it reduces to the binomial coefficient, for example 4! / (2! 2!) = 6.
- It gives the coefficient of each term when (x1 + x2 + ... )^n is expanded.
- It is the counting factor in the multinomial probability distribution.
- Group sizes must be whole numbers; decimals are rejected as invalid.
Multinomial coefficient: frequently asked questions
What is a multinomial coefficient?
The multinomial coefficient counts the number of distinct ways to arrange n objects into ordered groups of sizes k1, k2, ..., km. It equals n! divided by the product k1! times k2! times ... times km!, where the group sizes must sum to n. It generalises the binomial coefficient to more than two groups.
What inputs does this calculator take?
Enter up to four group sizes (k1 through k4). The total n is computed automatically as their sum, and the coefficient n! divided by the product of the group factorials is returned. Leave unused group fields at 0.
How is this different from a binomial coefficient?
A binomial coefficient C(n,k) is the special case of the multinomial coefficient with exactly two groups of sizes k and n minus k. The multinomial coefficient allows any number of groups, which is why it appears when distributing items into more than two categories.
Where are multinomial coefficients used?
They count arrangements of a multiset (for example anagrams of a word with repeated letters), they give the coefficients in the expansion of a sum raised to a power, and they appear in the multinomial probability distribution for outcomes across several categories.
Are the results exact?
The multinomial coefficient is always a whole number and is computed from an exact mathematical definition. For very large group sizes the result can exceed the precision of double-precision arithmetic, in which case the displayed value is the nearest representable number.
Official sources
- NIST Digital Library of Mathematical Functions: Multinomials and multinomial coefficients.
- NIST/SEMATECH e-Handbook of Statistical Methods: Combinatorics.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.