# Multinomial Coefficient: Definition & Examples

A multinomial coefficient describes the number of possible partitions of n objects into k groups of size n1n2, …, nk.

The formula to calculate a multinomial coefficient is:

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

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.