Multinomial Coefficient

TL;DR

Counts how many ways objects can be partitioned into groups of given sizes.
Computed by dividing the factorial of the total number of items by the product of the factorials of each group size.
Commonly used in combinatorics and probability to count arrangements or outcome multiplicities.

Definition

The multinomial coefficient represents the number of ways to arrange objects into groups of specified sizes. For group sizes k1, k2, …, km that sum to n, the multinomial coefficient is

\binom{n}{k_1,k_2,\dots,k_m} = \frac{n!}{k_1!\,k_2!\cdots k_m!}

Explanation

The coefficient gives the count of distinct arrangements when objects are partitioned into m labeled groups of sizes k1 through km. It is obtained by taking the factorial of the total number of objects and dividing by the product of the factorials of each group’s size. The concept is widely used in combinatorics and in probability calculations involving multiple categories.

Examples

Arranging objects into groups

Suppose we have three different objects, A, B, and C, and we want to arrange them into groups of size 2, 3, and 4, respectively. The multinomial coefficient is written as:

(2, 3, 4) = \frac{(2+3+4)!}{2!3!4!}

According to the source, this evaluates to 1260 ways.

Probability with colored balls

Suppose a bag contains 5 red balls, 6 green balls, and 7 blue balls. To determine the probability of selecting 3 red balls, 4 green balls, and 5 blue balls from the bag, without replacement, the source gives the multinomial coefficient as:

(3, 4, 5) = \frac{(5+6+7)!}{5!6!7!}

The source states that this corresponds to a probability of 0.00217, or about 0.22%.

Use cases

Combinatorics: counting arrangements or partitions of objects into specified group sizes.
Probability: computing counts or probabilities for outcomes across multiple categories.

Combinatorics
Probability