# How to use the COMBINA function

**What is the COMBINA function?**

The COMBINA function calculates the number of combinations with repetition for a given number of elements from a larger group of elements. The COMBINA function was introduced in Excel 2013.

## 1. Introduction

**Explain what combination is?**

A combination is a way of selecting items from a collection where the order of selection does not matter.

For example, if you have three fruits, say an apple, an orange, and a pear. There are three combinations of two that can be drawn from this set:

- apple and a pear
- apple and an orange
- pear and an orange

**What is the name if the order of selection matter?**

Permutation. An example are digits in a phone number, the order is very important or you will call the wrong number.

**What is repetition allowed?**

Repetition allowed is when there can be duplicate values.

**What is an element?**

An element refers to an item which can be anything.

**What is the difference between the COMBIN function and the COMBINA function?**

The difference is that COMBIN counts the number of combinations without repetitions, while COMBINA counts the number of combinations with repetitions.

For example, if you have three items A, B, and C, and you want to choose two of them:

- COMBIN will give you three possible combinations: AB, AC, and BC.
- COMBINA will give you six possible combinations: AA, AB, AC, BB, BC, and CC.

**Explain what permutation is?**

A permutation is a way of arranging a set of elements in a specific order. For example, if you have three letters A, B, and C, you can arrange them in six different ways: ABC, ACB, BAC, BCA, CAB, and CBA.

Each of these arrangements is called a permutation of the three letters. The order of the objects matters in a permutation, so ABC and BAC are considered different permutations.

The PERMUT and PERMUTA function are in the Statistical category

**What is the formula behind the COMBINA function?**

nCr (with repetition) = (n+r-1)! / (r! * (n-1)!)

- n is the total number of items
- r is the number of items being chosen

**What is the difference between combination and permutation?**

- The order of the objects matters in a permutation, so ABC and BAC are considered different permutations.
- ABC and BAC are considered the same combination because the order doesn't matter.

## 2. Syntax

COMBINA(*number*, *number_chosen*)

number |
Required.Â A number greater than or equal to zero and greater than or equal to Number_chosen. |

number_chosen |
Required. A number greater than or equal to 0. |

## 3. Example 1

**How many different combinations are possible if you can pick two items from a group of three different items? Repetition is allowed.**

Here are the arguments:

- number: 3
- number_chosen: 2

Formula in cell D3:

The formula returns 6 different combinations. If we name each item from 1 to 3 we can display the different possible combinations based on 2 items out of 3 with repetition allowed.

Cell range C13:C18 shows these different combinations in the image above, they are:

n | Combination (repetition) |

#1 | 11 |

#2 | 12 |

#3 | 13 |

#4 | 22 |

#5 | 23 |

#6 | 33 |

The math formula behind the COMBINA function is:

Combinations (with repetition) = (n+r-1)! / (r! * (n-1)!)

- n is the total number of items. In this example: 3
- r is the number of items being chosen. In this example: 2

Lets use the provided values in the question to calculate the number of combinations manually.

(n+r-1)! / (r! * (n-1)!)

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

=4!/4

=4*3*2*1/4

=24/4

=6

The math formula calculates the number of combinations to 6 which matches the value in cell C6.

## 4. Example 2

**You have 4 different types of fruits and want to make a fruit salad with 3 fruits. If you can use the same fruit more than once, how many different combinations are possible?**

Here are the arguments:

- number: 4
- number_chosen: 3

Formula in cell D3:

The formula returns 20 different combinations of fruits in the salad, repetition is allowed.

The math formula behind the COMBINA function is:

Combinations (with repetition) = (n+r-1)! / (r! * (n-1)!)

- n is the total number of items. In this example: 4
- r is the number of items being chosen. In this example: 3

Lets use the provided values in the question to calculate the number of combinations manually.

(n+r-1)! / (r! * (n-1)!)

=(4+3-1)! / (3!*(4-1)!)

=6!/(6*6)

=6*5*4*3*2*1/36

=720/36

=20

The math formula calculates the number of combinations to 20 which matches the value in cell C6.

## 5. Example 3

**In a game, you need to pick 4 numbers from 1 to 6. Numbers can be repeated. How many different combinations are possible?**

Here are the arguments:

- number: 6
- number_chosen: 4

Formula in cell D3:

The formula returns 126 different combinations of numbers, repetition is allowed.

The math formula behind the COMBINA function is:

Combinations (with repetition) = (n+r-1)! / (r! * (n-1)!)

- n is the total number of items. In this example: 6
- r is the number of items being chosen. In this example: 4

Lets use the provided values in the question to calculate the number of combinations manually.

(n+r-1)! / (r! * (n-1)!)

=(6+4-1)! / (4!*(6-1)!)

=9!/(4!*5!)

=(9*8*7*6*5*4*3*2*1)/(4*3*2*1*5*4*3*2*1)

=362880/2880

=121

The math formula calculates the number of combinations to 121 which matches the value in cell C6.

## 6. Example 4

**A restaurant offers 5 different toppings for pizza. If you want to order a pizza with 3 toppings, in how many different ways can you choose the toppings if the order of the toppings doesn't matter?**

Here are the arguments:

- number: 5
- number_chosen: 3

Formula in cell D3:

The formula returns 35 different combinations of numbers, repetition is allowed.

The math formula behind the COMBINA function is:

Combinations (with repetition) = (n+r-1)! / (r! * (n-1)!)

- n is the total number of items. In this example: 5
- r is the number of items being chosen. In this example: 3

Lets use the provided values in the question to calculate the number of combinations manually.

(n+r-1)! / (r! * (n-1)!)

=(5+3-1)! / (3!*(5-1)!)

=7!/(3*2*4*3*2)

=(7*6*5*4*3*2*1)/()

=5040/144

=35

The math formula calculates the number of combinations to 35 which also matches the value in cell C6.

### Functions in 'Math and trigonometry' category

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

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