Difference between Permutation and Combination

Difference between Permutation and Combination

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Vikram
Vikram Singh
Assistant Manager - Content
Updated on Mar 6, 2023 15:35 IST

In mathematics, Permutation and Combination are one of the most confusing topics as they are closely related. Permutation counts the number of different arrangements from n objects; on the other hand, the Combination is simply counting the number of ways an object is selected from n objects. In this article, we will briefly discuss the difference between Permutation and Combination.

permutation and combination

 

Permutation and Combination is an arrangement of a certain number of items. It is mainly concerned with determining the number of different ways of arranging and selecting objects from a given number of objects without actually listing them. In this article, we will discuss the difference between Permutation and Combination. But before moving forward, let’s discuss the Fundamental Principle of Counting.

Fundamental Principal Counting is a rule to count the total number of possible outcomes in a situation, i.e., if there are two events, one occurs in different ways, and another event occurs in n different ways, then the total number of occurrences of the event is x n.

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Table of Content

Permutation vs. Combination: Difference between Permutation and Combination

Parameter Permutation Combination
Definition Permutation is an arrangement of all members in order. A combination is a selection of members from a collection or group.
Represents Arrangement Selection
Order Values are arranged in an order. Values are not arranged in any specific order.
Derivation Multiple permutations from a single combination. Single combination from a single permutation.
Formula nPr = n!/(n-r)! nCr = n!/(r!)(n-r)!

What is Permutation?

Definition

A mathematical operation used to generate all the possible combinations of a set is called a permutation. In other words, Permutation is a method of arranging some or all the members of a set in a specific order

  • It implies all the possible arrangements or rearrangements of a set into different orders.
  • There are two types of Permutation:
    • Permutations with repetition
      • The number of permutations of n objects taken all at a time is nn.
      • The number of permutations of n objects taken r at a time is nr.
    • Permutations without repetitions

Formula of Permutation

nPr = n! / (n-r)!, where

  • n: n is the number of different elements
  • r: r is the arrangement pattern of the elements 
  • n, r are positive integers

Example of Permutation

Example 1In how many ways can the letter of the word “Naukri” be arranged so that all the vowels come together?

Answer 1: The word “naukri” has six letters and three vowels – ‘a’, ‘u’, and ‘i’ and all these vowels should come together. So these vowels can be clubbed together as a single letter, and the word naukri can be written as nkr(aui). i.e., there are only four letters left.

Hence, the number of ways to arrange these four letters are:

= 4! = 4 x 3 x 2 x 1 = 24,

and since all the three vowels (aui) are different among themselves, so

The number of ways to arrange these vowels among themselves is:

= 3! = 3 x 2 x 1 = 6

Hence, the number of ways can the letter of the word “naukri” be arranged so that all the vowels come together are:

= 24 x 6 = 144

Example 2: 3-letter words (with or without meaning) that can be formed out of the letter ‘Naukri’ if repetition is not allowed.

Answer 2: The word ‘naukri’ has six different letters. Hence, the 3-letter word (with or without meaning) formed by using these letters are:

= 6P3 = (6!)/(6-3)! = 6 x 5 x 4 x 3! / 3! = 6 x 5 x 4 = 120.

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What is a Combination?

Definition

The Combination is the different ways of selecting a group by taking only some members of the set without following any order. 

  • The Combination is used when we are only interested in selecting r objects from the set of n objects despite arranging all the items of the set.
  • It is all about grouping; the number of different groups of objects that can be formed from the available objects is calculated using the Combination.

Formula of Combination

nCr = nPr / r! = n! / r!(n-r)!, where

  • n: n is the number of different elements
  • r: r is the collection of the elements 
  • n, r are positive integers

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Example of Combination

Example 3: In how many ways can 7-kabaddi players be selected out of a total of 10 players, such that two particular players (captain and vice-captain) should be included in each team?

Answer 3: Since we have to add the captain and vice-captain to each team.

Therefore, we have to select the remaining (7-2) = 5 players from (10-2) = 8 players.

Hence, the required number of ways 7-kabaddi players can be selected out of a total of 10 players, such that two particular players should be included in each team:-

= 8C5 = (8!) / (5!)(8-5)! = 8 x 7 x 6 x 5! / (5!) x (3!) = 8 x 7 x 6 / 3 x 2 x 1 = 8 x 7 = 56.

Example 4: Find the number of triangles that can be drawn out of n given points on a circle.

Answer 4: Since to make a triangle, we only need three points out of n points on the circle; therefore, the number of triangles = the number of ways of choosing 3 points from n points
= nC3 = (n!) / (3!) (n-3)! = n x n – 1 x n – 2 x (n -3)! / 3! x (n-3) = n(n-1)(n-2)/6.

Application of Permutations and Combinations

  • Allocation of Phone Numbers/City Code/Country Telephone Code/Car Numbers
  • Computer Architecture: The design of computer chips involves consideration of possible Permutation of inputs and output pins.
  • Machine Learning: Computer languages are closely related to combinatorics. This includes String Search Algorithms, word processing, and databases.
  • Database and Data Mining: Queries in the database are multiple join operations that are permutations of the constituent join operation, so determining the optimal permutation combination that gives the minimum cost becomes a significant problem.
  • Operational Research: In operational research, you will find problems like job-assigning, minimizing total time and cost, and maximizing the revenue, i.e., the problems in which you have to optimize variables based on different constraints. All these problems are exceptional cases of combinatorics.
  • Simulation: Permutations and combinations are widely used for simulation in different areas, such as genetics, networking, cryptography, and others.

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A key Difference Between Permutation and Combination

  • A permutation is used when the problem asks for the number of arrangements and different objects to be counted. In contrast, the Combination is used when the problem asks for the number of ways of selecting objects, and the selection order is not to be counted. 
  • The fundamental difference between Permutation and Combination is order, placement, and position.
    • In Permutation, position, placement, and order matter.
    • In Combination, order, placement, and position don’t matter.
  • Permutation denotes different ways of arranging digits, alphabets, colours, etc., while Combination refers to the several ways of selecting food items, clothes, etc.
  • A permutation is done in a sequence, whereas a Combination is done with any rotation.
  • Permutation can happen only in groups, and once you have done the Permutation, it can’t be reversed, while in the Combination, you can choose one or more items from the same group.

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Conclusion

In this article, we have discussed the difference between Permutation and Combination, their example and application.

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About the Author
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Vikram Singh
Assistant Manager - Content

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio

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