# How to use the MULTINOMIAL function

**What is the MULTINOMIAL function?**

The MULTINOMIAL function calculates the ratio of the factorial of a sum of values to the product of factorials.

#### Table of Contents

## 1. Introduction

**What is the sum of values?**

Summarize is when you add numbers using addition and the result is the sum or total.

For example, lets use the numbers 2, 3 and 4.

2 + 3 + 4 = 9

The sum is 9.

**What is the factorial of a sum of values?**

The factorial of a number n is the product of all positive integers less than or equal to n, and is represented by n!

The factorial of a number n is defined as:

n! = n * (n-1) * (n-2) * ... * 3 * 2 * 1

In other words, the factorial of a number n is the product of all positive integers less than or equal to n.

Some examples:

1! = 1

2! = 2 * 1 = 2

3! = 3 * 2 * 1 = 6

4! = 4 * 3 * 2 * 1 = 24

5! = 5 * 4 * 3 * 2 * 1 = 120

0! = 1

The factorial function grows very quickly as n increases. Factorials are useful in combinatorics for counting permutations and combinations.

Using the example that returned the sum of 9, the factorial becomes 9!

9*8*7*6*5*4*3*2*1 = 362880

**What is the product of values?**

Multiplication is one of the basic arithmetic operations, along with addition, subtraction, and division. Multiplication involves taking two numbers, called factors or terms, and combining them to get a product.

It is denoted by the multiplication symbol * or x. For example: 5 * 3 = 15

**What is the factorial of a product of values?**

The factorial of a product is equal to the product of the individual factorials.

For example:

(2*3*4)! = 2! * 3! * 4! = 2 * 6 * 24= 288

**What is the ratio of the factorial of a sum of values to the product of factorials?**

multinomial coefficient = (a_{1} + a_{2} + ... + a_{n})! / (a_{1}! a_{2}! ... a_{n}!)

Proceeding with the values above 2,3, and 4 we get

362880 / 288 equals 1260.

**What is the ratio?**

A ratio shows the relative size of two or more values. It is one number divided by another. A ratio compares two quantities using division.

Ratios can be written as fractions, with the numerator and denominator representing the values being compared. Use / in an Excel formula to perform a division and create a ratio.

**What functions use the factorial in Excel?**

Function | Description |
---|---|

MULTINOMIAL(n1, n2, ...) |
Returns the ratio of the factorial of a sum of values to the product of factorials. |

FACT(number) |
Returns the factorial of the provided number. |

FACTDOUBLE(number) |
Returns the double factorial of the provided number. |

## 2. Syntax

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

number1 |
Required. The number for which you want to calculate the multinomial. |

[number2] |
Optional. Up to 254 additional numbers. |

## 3. Example 1

Cells B3:D3 contains the following numbers 2, 3, and 4. The function in cell F3 calculates the multinomial to 1260.

Formula in cell D3:

Lets calculate the multinomial and verify the value in cell F3. The math formula is:

The text representation of the math formula is: multinomial coefficient = (a_{1} + a_{2} + ... + a_{n})! / (a_{1}! a_{2}! ... a_{n}!)

=(2+3+4)! / (2!*3!*4!)

=9!/4*3*2*3*2*2

=362880/288

=1260

1260 matches the calculated value in cell F3.

The following Excel formula calculates the same thing as the MULTINOMIAL function:

The second group of values are 2, 4, and 5 in cell range B4:D4.

Formula in cell D4:

The calculated value is 6930, lets calculate this value manually.

=(2+4+5)!/(2!*4!*5!)

=11!/2*4*3*2*5*4*3*2

=39916800/5760

=6930

This value matches the calculated value in cell D4.

The third group of values are 2, 5, and 7 in cell range B5:D5.

Formula in cell D4:

The calculated value is 72072, lets calculate this value manually.

=(2+5+7)!/(2!*5!*7!)

=14!/2*5*4*3*2*7*6*5*4*3*2

=87178291200/1209600

=72072

This value matches the calculated value in cell D5.

## 4. Example 2

**In a team of 10 people, in how many ways can you choose 3 people to be in the first group [A], 2 people to in the second group [B], and the remaining 5 people to be in the third group [C]?**

The arguments are:

- Cell B3 - 3
- Cell C3 - 2
- Cell D3 - 5

Formula in cell F3:

The formula returns 2520 which represents the number of ways you can choose 3 people to be in the first group, 2 people in the second group, and 5 people in the third group.

Lets calculate this manually. multinomial coefficient = (a_{1} + a_{2} + ... + a_{n})! / (a_{1}! a_{2}! ... a_{n}!)

=(3+2+5)!/(3!*2!*5!)

=3628800/1440

=2520

This value matches the value in cell F3.

## 5. Example 3

**In how many ways can you arrange the letters in the word "ANTARCTICA"?**

The count for each letter is: A = 3, N = 1, T = 2, R = 1, I = 1, C = 2

The arguments are populated in the following cells:

- Cell B3 - 3
- Cell C3 - 1
- Cell D3 - 2
- Cell B4 - 1
- Cell C4 - 1
- Cell D4 - 2

Formula in cell F3:

The formula returns 151200 which represents the number of ways you can arrange the letters in the word "ANTARCTICA".

Lets calculate this manually. multinomial coefficient = (a_{1} + a_{2} + ... + a_{n})! / (a_{1}! a_{2}! ... a_{n}!)

=(3+1+2+1+1+2)!/(3!*2!*2!)

=3628800/24

=151200

This value matches the value in cell F3.

## 6. Example 4

**A machine produces 20% defective items. What is the probability of getting exactly 3 defective items out of 10 items?**

The arguments are populated in the following cells:

- Cell B3 - 3
- Cell C3 - 7

Formula in cell F3:

The formula returns 0.201326592 (20.1%) which represents the probability of getting exactly 3 defective items out of 10 items based on an 20% defective output.

Lets calculate this manually. multinomial coefficient = (a_{1} + a_{2} + ... + a_{n})! / (a_{1}! a_{2}! ... a_{n}!)

=(3+7)!/(3!*7!)

=3628800/30240

=120

=120*0.2^3*0.8^7

=120*0.008*0.2097152

=0.201326592

This value matches the value in cell F3.

## 7. Example 5

**In a lottery where you pick 6 numbers from 1 to 49, how many possible combinations are there?**

The arguments are populated in the following cells:

- Cell B3 : 6
- Cell C3 : 49 - 6 = 43

Formula in cell F3:

The formula returns 13,983,816 which represents the number of possible combinations if you pick 6 numbers from 49 numbers (1 to 49).

_{1} + a_{2} + ... + a_{n})! / (a_{1}! a_{2}! ... a_{n}!)

=(6+43)!/(6!*43!)

=49!/(6!*43!)

=13,983,816

This value matches the value in cell F3.

## 8. The function not working

MULTINOMIAL returns

- #VALUE! if argument is nonumeric.
- #NUM! if argument is less than 0 (zero).

### Functions in 'Math and trigonometry' category

The MULTINOMIAL function function is one of 61 functions in the 'Math and trigonometry' category.

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