Multinomial Coefficient Calculator
Compute the multinomial coefficient n!/(k₁!k₂!…kₘ!) exactly for any group sizes, or count the distinct arrangements of the letters of a word.
The multinomial coefficient counts the ways to split n distinct items into labelled groups of given sizes k₁, k₂, …, kₘ. The same number counts the distinct arrangements of a word with repeated letters, and gives the coefficients in the expansion of (x₁ + x₂ + … + xₘ)ⁿ. This calculator uses exact whole-number arithmetic, so even answers with hundreds of digits are given digit for digit.
How to use this tool
- Choose group sizes, or a word whose letters you want to rearrange.
- Enter the sizes separated by commas (for example 3, 2, 2), or type the word.
- Read the coefficient; n is the sum of the group sizes.
- The table rebuilds the answer as a chain of binomial coefficients.
Formula
(n; k₁, k₂, …, kₘ) = n! ÷ (k₁! × k₂! × … × kₘ!), n = k₁ + k₂ + … + kₘ
Equivalently it is a product of binomial coefficients, C(n, k₁) × C(n − k₁, k₂) × …: choose the members of the first group, then the second group from those left, and so on. With two groups it reduces to the ordinary binomial coefficient C(n, k), and 0! = 1, so an empty group changes nothing.
Example
The word MISSISSIPPI has 11 letters: M once, I four times, S four times and P twice. The number of distinct arrangements is 11! ÷ (1! × 4! × 4! × 2!) = 39,916,800 ÷ 1,152 = 34,650.
Frequently asked questions
How is this different from a combination?
A combination C(n, k) splits items into two groups, chosen and not chosen. The multinomial coefficient allows any number of groups: splitting 7 people into a labelled team of 3 and two labelled pairs gives 7! ÷ (3! × 2! × 2!) = 210 ways.
What does it have to do with (a + b + c)ⁿ?
In the expansion of (a + b + c)⁷, the coefficient of a³b²c² is 7! ÷ (3! × 2! × 2!) = 210. The multinomial theorem says this holds for any number of terms and any powers that add up to n.
What if the groups are not labelled?
If several groups have the same size and their order does not matter, divide by the factorial of how many such groups there are. Splitting 6 people into three unlabelled pairs gives 6! ÷ (2! × 2! × 2!) ÷ 3! = 90 ÷ 6 = 15.
How large can the numbers get?
The total n can be up to 1,000. Results are exact; numbers with more than 15 digits are also shown in scientific notation.
Last updated: 2026-10-30