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 = (a1 + a2 + ... + an)! / (a1! a2! ... an!)
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 = (a1 + a2 + ... + an)! / (a1! a2! ... an!)
=(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 = (a1 + a2 + ... + an)! / (a1! a2! ... an!)
=(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 = (a1 + a2 + ... + an)! / (a1! a2! ... an!)
=(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 = (a1 + a2 + ... + an)! / (a1! a2! ... an!)
=(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).
Lets calculate this manually. multinomial coefficient = (a1 + a2 + ... + an)! / (a1! a2! ... an!)
=(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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