MULTINOMIAL function

The MULTINOMIAL function in Excel returns the multinomial coefficient of a set of numbers. This is used to calculate the coefficient of a multinomial expression, which is a generalization of the binomial coefficient. The multinomial coefficient is the number of ways to arrange a set of objects where some objects may be repeated.

Syntax

=MULTINOMIAL(number1, [number2], ...)

Parameters

• number1, number2, ...: These are the numbers or ranges that represent the group sizes. Excel can accept up to 30 arguments. Each number represents a part of the multinomial coefficient formula.

Formula Explanation

The MULTINOMIAL function calculates the coefficient of the multinomial expansion using the formula:

$$\frac{(n!)}{(x_1! \cdot x_2! \cdot … \cdot x_k!)}$$

Where:

• n is the total sum of the numbers.
• x₁, x₂, …, xk are the individual numbers.

In other words, it calculates how many ways you can arrange a multiset of objects.

Example

If you want to calculate the number of ways to arrange 5 objects of type A, 3 objects of type B, and 2 objects of type C, you can use the MULTINOMIAL function.

Formula:

=MULTINOMIAL(5, 3, 2)

Calculation:

The result will be calculated as:

$$\frac{(5 + 3 + 2)!}{5! \cdot 3! \cdot 2!} = \frac{10!}{5! \cdot 3! \cdot 2!} = \frac{3,628,800}{120 \cdot 6 \cdot 2} = 252$$

Result:

252

Explanation:

• The function calculates the number of ways to arrange 5 objects of type A, 3 objects of type B, and 2 objects of type C, which equals 252 possible arrangements.

Real-World Use Case:

This function is particularly useful in probability and combinatorics, especially when dealing with problems involving distributing indistinguishable objects into distinguishable bins or classes, such as calculating the likelihood of specific outcomes in a multinomial distribution or determining the number of ways to arrange repeated items in a set.

Key Notes:

• All the numbers passed to the MULTINOMIAL function must be non-negative integers.
• The function can accept up to 30 arguments.
• The MULTINOMIAL function calculates the multinomial coefficient but does not directly compute probabilities or apply specific distribution formulas.

Common Applications

• Probabilistic Models: In statistics, particularly when working with multinomial distributions.
• Combinatorics: When determining the number of ways to arrange items in a set where there are repetitions.
• Game Theory: For calculating the number of possible outcomes in games or simulations where multiple categories of items exist.