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What is the Combination and Combination Formula? How to work with the Combination Formula?


In this article, you will learn what combination is and how it is calculated. You will also know the relationship between permutation and combination.


What is Combination?

In mathematics, the combination is a technique that determines the number of possible arrangements for data collection. In combination, any number of arrangements for the collection of data can be selected. Unlike permutation, the order of the selection of objects does not matter in combination.

The combination is a mathematical term that is also important in statistics because it is used to select data samples. It can also be used in finance and other disciplines.

Combination Formula

The combination is used to find the possible number of arrangements for a collection of data sets without considering the order of the arrangements. For example in the arrangement of AB and BA, both are considered as a single arrangement ignoring the order. The formula is used to calculated combination for a collection of data:

Combination Formula

Combination Formula


It can also be denote by Cn,r


For example, if the 3 fruits out of 5 fruits are selected then the number of possible combination are

$$ ^nC_r \;=\; \frac{n!}{r!(n-r)!} \;=\; {5!}{3!(5-3)!} $$ $$ ^nC_r\;=\; \frac{5!}{(3!×2!)} \;=\; {5×4×3!}{(3!×2!)} \;=\; {20}{2} \;=\; 10 $$

So the 10 possible arrangements are possible with the 3 fruits selected.

How does the Combination Formula Work?

Let a collection of data has n number of elements. If r number of elements are selected form the collection. Then the combination formula without repetition is:

$$ ^nC_r \;=\; (n r ) \;=\; \frac{n!}{r!(n-r)!} $$

The number of possible arrangements for the ‘r’ number of selected elements can be calculated by using the above formula. Let’s see the following examples to understand how the combination is calculated.

Calculate combination

Calculate combination


Permutation and Combination Formula


The permutation and combination are almost the same terms but the difference is that the order of arrangements matters in permutation. In permutation, the elements are arranged in a particular way. The permutation formula is given by:

$$ ^nP_r \;=\; \frac{n!}{(n-r)!} $$

Whereas the combination formula is:

$$ C_{n,r} \;=\; \frac{n!}{r!( n-r)!} $$

Representation of Combination:

The combination can be denoted in the following ways:

$$ ^nC_r \;=\; \frac{n!}{r!(n-r)!} $$ $$ C_{n,r} \;=\; \frac{n!}{r!(n-r)!} $$ $$ C(n,r) \;=\; \frac{n!}{r!(n-r)!} $$


What is the Formula for Combination nCr?

The formula of combination for n number of elements with r number of selected elements is:

$$ C_{n,r} \;=\; \frac{n!}{r!(n-r)!} $$

What is Permutation and Combination?

The permutation is the arrangement of elements in a particular way according to the order of the elements. And the combination is a technique of arranging elements in all possible ways that you may calculate by using permutations and combinations calculator.

What is the Combination in Real Life?

The common examples of combination in real life is the selection of menu, food, clothes, subjects and team etc.

Alan Walker

Last updated: July 07, 2023

Studies mathematics sciences, and Technology. Tech geek and a content writer. Wikipedia addict who wants to know everything. Loves traveling, nature, reading. Math and Technology have done their part, and now it's the time for us to get benefits.