Multinomial Coefficient: Definition & Examples

A multinomial coefficient describes the number of possible partitions of n objects into k groups of size n₁, n₂, …, n_k.

The formula to calculate a multinomial coefficient is:

Multinomial Coefficient = n! / (n₁! * n₂! * … * n_k!)

The following examples illustrate how to calculate the multinomial coefficient in practice.

Example 1: Letters in a Word

How many unique partitions of the word ARKANSAS are there?

Solution: We can simply plug in the following values into the formula for the multinomial coefficient:

n (total letters): 8

n₁ (letter “A”): 3

n₂ (letter “R”): 1

n₃ (letter “K”): 1

n₄ (letter “N”): 1

n₅ (letter “S”): 2

Multinomial Coefficient = 8! / (3! * 1! * 1! * 1! * 2!) = 3,360

There are 3,360 unique partitions of the word ARKANSAS.

Example 2: Students by Grade

A group of six students consists of 3 seniors, 2 juniors, and 1 sophomore. How many unique partitions of this group of students are there by grade?

Solution: We can simply plug in the following values into the formula for the multinomial coefficient:

n (total students): 6

n₁ (total seniors): 3

n₂ (total juniors): 2

n₃ (total sophomores): 1

Multinomial Coefficient = 6! / (3! * 2! * 1!) = 60

There are 60 unique partitions of these students by grade.

Example 3: Political Party Preference

Out of a group of ten residents in a certain county, 3 are Republicans, 5 are Democrats, and 2 are Independents. How many unique partitions of this group of residents are there by political party?

Solution: We can simply plug in the following values into the formula for the multinomial coefficient:

n (total residents): 10

n₁ (total Republicans): 3

n₂ (total Democrats): 5

n₃ (total Independents): 2

Multinomial Coefficient = 10! / (3! * 5! * 2!) = 2,520

There are 2,520 unique partitions of these residents by political party.

Additional Resources

The multinomial coefficient is used in part of the formula for the multinomial distribution, which describes the probability of obtaining a specific number of counts for k different outcomes, when each outcome has a fixed probability of occurring.

Bonus: You can use the Multinomial Coefficient Calculator to easily calculate multinomial coefficients.