*53*

A **multinomial coefficient** describes the number of possible partitions of *n* objects into *k* groups of size *n _{1}*,

*n*, …,

_{2}*n*.

_{k}The formula to calculate a multinomial coefficient is:

Multinomial Coefficient = n! / (n_{1}! * n_{2}! * … * 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

**n _{1}** (letter “A”): 3

**n _{2}** (letter “R”): 1

**n _{3}** (letter “K”): 1

**n _{4}** (letter “N”): 1

**n _{5}** (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

**n _{1}** (total seniors): 3

**n _{2}** (total juniors): 2

**n _{3}** (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

**n _{1}** (total Republicans): 3

**n _{2}** (total Democrats): 5

**n _{3}** (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.