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Difference Between Permutation and Combination

Permutation and combination are essential mathematical concepts applied in other science areas as well. Since they are similar ideas, they are frequently confused with one another and used interchangeably without realizing it. However, a clear concept of both mathematical ways is required for its proper application.

The main difference between permutation and combination is that “permutation’’ is an ordered combination, whereas "combination" refers to any choice or pairing of values within a particular set of criteria or classification. 

Combinations emphasize choice more than order, placement, or organization. Values come in both solitary and plural forms. The three characteristics mentioned above are widely prized in permutations, nevertheless. In addition to these three, a permutation also shows the location of each value. In one statement, we can say the permutation follows an order, but the combination does not.

In this article, we will shed more light on permutation and combination, which will clarify the differences between these two.

What is a Permutation?

Permutation Formula

Suppose we have some data, and we want to make every possible arrangement from the data. All these possible arrangements are called permutations. 

For example, we have three things, a, b, and c, so all the possible permutations of these letters, taking two at a time, will be AB, BA, BC, CB, AC, CA. So, the number of permutations of three things taken two at a time is 6. Other permutations are also possible using three things at a time, such as ABC, ACB, CAB, etc., which results in 6 permutations altogether.

We will further discuss a few rules and theorems used to calculate the number of possible permutations. Do note that in permutations, the arrangement order is considered. When the order of arrangement is changed, a different permutation is obtained.

What is a Combination? 

Combination Formula

Each of the different selections made by using some or all of the objects given, irrespective of their arrangement, is called a combination. 

For example, the list of combinations formed of three letters A, B, and C, taken two at a time, are AB, AC, and BC.  Likewise, the combination of four letters A, B, C, and D, taken two at a time, will be AB, AC, AD, BC, BD, and CD.

Note that AB and BA will be the same combination and will count as one.

Key Differences between Permutations and Combinations

Differences in regards to the order

A permutation is an ordered combination, whereas "combination" refers to any choice or pairing of values within a particular set of criteria or classification. In permutation, a selection and an arrangement are made, but in combination, only the selection is essential.

Differences in the number of results

A single combination can lead to a variety of permutations. Nevertheless, only one arrangement is required for one permutation. For example, from a combination of AB, the permutations obtained are AB and BA. Hence, AB and BA are the same combinations but with different permutations. Hence, the number of results from the same set of data changes.

Differences in the number of permutations exceeding the number of combinations

To calculate the number of permutations of 'n’ different items or things, taken ‘r’ at a time, we take ‘r’ items from the available 'n’ items and then make the required arrangement. So, the number of resultant permutations becomes more significant than the number of possible combinations. Each combination gives many permutations. For example, the six permutations ABC, ACB, BCA, BAC, CBA, and CAB are from the same combination of ABC.

Difference in representation

The number of combinations of n objects that are taken ‘r’ at a time is represented by C(n,r) or nCr, whereas permutation has the formula P(n, r) or nPr.

Comparison Chart: Permutation Vs Combination

ParametersPermutationCombination
DescriptionIt is defined as arranging ‘r’ objects from a given set of ‘n’ objectsIt is defined as grouping ‘r’ objects from a given set of ‘n’ objects
Order Order matters hereThe order does not matter here
NotationDenoted by nPrDenoted by nCr
RepresentationIt represents an arrangementRepresents grouping

Similarities between Permutations and Combinations

The topics of permutation and combination are pretty similar, but they are not interchangeable. The listed differences indicate the heterogeneity of the two mathematical concepts. The concepts and formulas are different, as are the outcomes. But, when inspected closely, we can point out a few similarities between the two, like:

  • In both cases, repetition of an order is not possible.
  • They are both used for solving mathematical problems. 
  • Both play an essential role in probabilities and statistics.
  • We select the ‘r’ number of items from a given ‘n’ set of things.

FAQs

When are permutations and combinations used?

A permutation is used for a list of data, in which the data sequence is important. In comparison, for a group of data, where the sequence of the data is irrelevant, a combination is used.

When to use permutation and when to use combination?

Students frequently have trouble determining if a given task calls for combinations or permutations. First, decide whether or not the order has to be taken into account while tallying the alternatives. Use permutations if the order is essential and choices cannot be repeated (using the formula for permutation). Use combinations if the order is unimportant and the choices cannot be repeated (using the combinations formula).

How are permutations and combinations used in our daily lives?

We can use permutations to find the different number of ways competitors will finish a race. For example, if 10 people are running a marathon race, we can use factorials to calculate the number.

How are permutation and combination related?

Both are mathematical concepts required for solving probability problems.

Conclusion

The difference between permutation and combination can easily be understood from the above description and can be used for day-to-day applications.

References

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About the Author: Nicolas Seignette

Nicolas Seignette, who holds a scientific baccalaureate, began his studies in mathematics and computer science applied to human and social sciences (MIASHS). He then continued his university studies with a DEUST WMI (Webmaster and Internet professions) at the University of Limoges before finishing his course with a professional license specialized in the IT professions. On 10Differences, he is in charge of the research and the writing of the articles concerning technology, sciences and mathematics.
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