## Definition

Permutation and combination are the ways to represent a group of objects by selecting them in a set and forming subsets. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination.2

### Permutation

A permutation is the choice of r things from a set of n things without replacement, where order matters, and denoted by:

In mathematics, permutation relates to the act of arranging all the members of a set into some sequence or order. In other words, if the set is already ordered, then the rearranging of its elements is called the process of permuting. Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered.2

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word “permutation” also refers to the act or process of changing the linear order of an ordered set.5

### Combination

A combination is the choice of r things from a set of n things without replacement, where the order does not matter, and denoted by:

The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection or k-combination with repetition are often used.2

## What

Permutations are used in almost every branch of mathematics, and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences.

Permutations are used in the interleaver component of the error detection and correction algorithms, such as turbo codes, for example 3GPP Long Term Evolution mobile telecommunication standard uses these ideas (see 3GPP technical specification 36.212[58]). Such applications raise the question of fast generation of permutations satisfying certain desirable properties. One of the methods is based on the permutation polynomials. Also as a base for optimal hashing in Unique Permutation Hashing.5

## Why

For example, permutations and combinations is very useful in computer science. It is prerequisite to Graph theory, Probability, etc. Discrete mathematics is one of the most important course in computer science and permutations and combinations is must to learn that course. There are so many applications to graph theory even this Quora or Facebook uses many concepts of graph theory. Probability and permutations and combinations are very much interlinked and are useful every where. More than anything it is also fun to learn and use permutations and combinations.4

## How

In this article, we are going to learn how to calculate the number of ways in which x balls can be distributed in n boxes. This is one confusing topic which is hardly understood by students. But once mastered, it is the easiest topic of Permutation and Combination.1

### Types of Permutations

Permutation can be classified in three different categories: 5

1. Permutation of n different objects (when repetition is not allowed)
2. Repetition, where repetition is allowed
3. Permutation when the objects are not distinct (Permutation of multi sets)

Many of the References and Additional Reading websites and Videos will assist you with understanding and using combinations and permutations.

As some professors say: “It is intuitively obvious to even the most casual observer.

## References

1 “Permutation And Combination Questions – Distribution Of Balls – hitbullseye “. 2022. hitbullseye.com. https://www.hitbullseye.com/Quant/Permutation-and-Combination-Concept.php.

2 “Permutation And Combination (Definition, Formulas And Examples)”. 2022. BYJUS. https://byjus.com/maths/permutation-and-combination/.

4 “What Is The Use Of Combinations And Permutations In Computer Science?” 2022. Quora. https://www.quora.com/What-is-the-use-of-combinations-and-permutations-in-computer-science/answer/Raviteja-Kakarala.

5 Jogdand, Aryan. “PERMUTATIONS”. 2022. Medium. https://medium.com/@aryanjogdand18rn/permutations-a0b216dcb462.

“A Complete Guide To Permutations And Combinations – Lambda Geeks”. 2020. Lambda Geeks. https://lambdageeks.com/permutations-and-combinations/.

Permutations and Combinations, this article will discuss the concept of determining, in addition to the direct calculation, the number of possible outcomes of a particular event or the number of set items, permutations and combinations that are the primary method of calculation in combinatorial analysis.

“Easy Permutations And Combinations – BetterExplained”. 2022. betterexplained.Com. https://betterexplained.com/articles/easy-permutations-and-combinations/.

“Permutation And Combination – GeeksForGeeks”. 2016. GeeksForGeeks. https://www.geeksforgeeks.org/permutation-and-combination/.

“Permutation And Combination | Set-2 – GeeksForGeeks”. 2019. GeeksForGeeks. https://www.geeksforgeeks.org/permutation-and-combination-set-2/.

“Permutation And Combination: Solved Examples, & Practice Questions”. 2018. toppr-guides. https://www.toppr.com/guides/quantitative-aptitude/permutation-and-combination/.

Permutation and Combination is a very important topic of mathematics as well as the quantitative aptitude section. Through permutations and combinations, we count the various arrangements that can be made from a certain group. Here we have all these concepts with a diverse set of solved examples and practice questions that will not only give you a perfect coverage of the syllabus but also help you solve any question in less than a minute.

Tech & Math. “4 Steps To Master Permutations And Combinations — Journey In Combinatorics”. 2022. Medium. https://techandmath.medium.com/4-steps-to-master-permutations-and-combinations-journey-in-combinatorics-f1f5ae201b28.

Well, combinatorics has been quite an interesting subject to talk about, and its ideas are widely applied to a wide range of area: computer science, physics, chemistry and etc. The article serves to help you understand permutations and combinations, two of the most fundamental ideas in a solid way by walking you through 4 steps:

Step 1. Understand multiplication rule with repetition
Step 2. Understand multiplication rule without repetition
Step 3. Understand multiplication rule when order does not matter
Step 4. Formalize our ideas into mathematical language

Tech & Math. “Simple Pigeonhole Principle, But Interesting Result — Journey In Combinatorics”. 2022. Medium. https://techandmath.medium.com/simple-pigeonhole-principle-but-interesting-result-journey-in-combinatorics-d0e120adbe37.

Pigeonhole principle is one of the most easy yet fundamental theorem in the area of discrete math. Nonetheless, if pigeonhole principle is applied appropriately to some scenarios, some amazing results that seemed pretty difficult to prove can be deduced. In this article, you will learn or refresh the pigeonhole principle with some of the classical application. Then, some nice problems involving pigeonhole principle technique will be discussed. After reading this, you should be able to appreciate this simple yet subtle theorem.

Pigeonhole principle states that if there are n pigeons to be allocated, but only n-1 cages, then there must exists a cage with at least 2 pigeons. The proof is quite straightforward: Assume the opposite, then we have at most 1 pigeon at each cage. Summing over then n-1 cage, we have the number of pigeons ≤ (n-1)*1= n-1. However, we have n pigeons. There must be a contradiction.

## Videos

This video tutorial focuses on permutations and combinations. It contains a few word problems including one associated with the fundamental counting principle. Permutations are useful to determine the different number of ways to arrange something where as combinations is useful for determining how many ways to combine something when the order does not matter such as selecting members to form a committee. In a permutation, the order matters. Examples include repeated symbols or arranging letters in a word such as Alabama or Mississippi. This video also discusses the basics of permutations and combinations using letters such as ABCD.

Introduction to combinations | Probability and Statistics | Khan Academy

Brett explains the difference between combinations and permutations and demonstrates how to solve a classic card counting problem both through reasoning and by applying the choose and permutate formulas. In today’s episode, you will learn when to use the combinations, or choose, formula and when to use the permutations formula as well as gain a deep understanding behind the formulas and combinatorics. You will also be introduced to factorial notation. We begin the lesson by solving, through reasoning, how many different 5-card hands can be drawn from a standard deck of 52 cards using slot diagrams. In the process we will uncover both formulas for permutations and combinations.

By popular demand, this is a video on everything you need to know about Permutations & Combinations. We’ll cover the basics of Permutations and Combinations, how to use the insertion method, as well as finding permutations in a circle. It is currently just the basics, but as we delve into more questions, we would see greater how the concepts here can be applied to those questions!

## What Is The Practical Real-Life Use Of Permutation And Combination?

### Application Areas of Combinatorics

Communication networks, cryptography and network security

Permutations are frequently used in communication networks and parallel and distributed systems . Routing different permutations on a network for performance evaluation is a common problem in these fields. Many communication networks require secure transfer of information, which drives development in cryptography and network security . This area has recently become particularly significant because of the increased use of internet information transfers. Associated problems include protecting the privacy of transactions and other confidential data transfers and preserving the network security from attacks by viruses and hackers. Encryption process involves manipulations of sequences of codes such as digits, characters, and words. Hence, they are closely related to combinatorics, possibly with intelligent encryption process. For example, one common type of encryption process is interchanging–i.e., permuting parts of a sequence . Permutations of fast Fourier transforms are employed in speech encryption

### Computer architecture

Design of computer chips involves consideration of possible permutations of input to output pins. Field-programmable interconnection chips provide user programmable interconnection for a desired permutation . Arrangement of logic gates is a basic element for computer architecture design

### Computational molecular biology

This field involves many types of combinatorial and sequencing problems such as atoms, molecules, DNAs, genes, and proteins One-dimensional sequencing problems are essentially permutation problems under certain constraints.

### Languages

Both natural and computer languages are closely related to combinatorics . This is because the components of these languages, such as sentences, paragraphs, programs, and blocks, are arrangements of smaller elements, such as words, characters, and atoms. For example, a string searching algorithm may rely on combinatorics of words and characters. Direct applications of this can include word processing and databases. Another important application area is performance analysis of these string searching algorithms. The study of computability–what we can compute and how it is accomplished–draws heavily on combinatorics.

### Pattern analysis

In a broad sense, all the above-mentioned areas can be viewed as special cases of pattern analysis. Molecular biology, for example, studies patterns of atoms, molecules, and DNAs whereas languages treat patterns of sentences, words, and strings. Patterns can have many other forms; for example, visual images, acoustic signals, and other physical quantities such as electrical, pressure, temperature, etc., that appear in engineering problems. Patterns can also be abstract without any associated physical meaning. These patterns may be represented in various ways such as digital, analog, and other units. Some of these types of patterns can be associated with combinatorics. There has been extensive research on combinatorial pattern matching (Combinatorial, 1992-2005). Computer music can be a specialized application domain of combinatorics of acoustic signals.

### Scientific discovery

For certain types of knowledge discovery problems, generation of combinatorial sequences may become necessary in the process of yielding candidate solutions. For example, in scientific discovery, we may want to have a sequence of plausible chemical/biological reactions and their formations . In each step of the sequence, we may generate combinatorial sequences of chemical/biological radicals, bases, and molecular compounds as candidate solutions and may select the most likely ones under certain rules and constraints. In another example, certain areas of mathematics, such as graph theory and number theory, may generate combinatorial sequences as candidate solutions.

### Databases and data mining

Queries in databases are multiple join operations that are permutations of the constituent join operations. Determining an optimal permutation that gives minimum cost is a common and important problem . Data mining or knowledge discovery in databases is a relatively new field that aims at distilling useful information, often from large databases. In this area, techniques employing symbolic AI can manipulate combinatorial sequences of atoms or information elements. Non-symbolic knowledge discovery techniques, such as genetic algorithms and genetic programming, most commonly deal with solutions in the form of sequences of bits, digits, characters, and sometimes Lisp program elements. Neural networks, another domain of non-symbolic AI, sometimes deal with combinatorial patterns. Knowledge discovery techniques under uncertainty, such as Bayesian networks, Dempster-Shafer theory, fuzzy logic, and rough set theory, may have combinatorial solutions

### Operations research

Many optimization problems in operations research (OR) involve combinatorics. The job scheduling problem is essentially a sequencing problem to determine the order of jobs to be processed in an effort to minimize the total time, cost, etc. Here, jobs can be in a computer system, network, or processing plant. Many problems involving graphs or networks also deal with the order of vertices and edges. The traveling salesperson problem is to determine the order of cities to be visited to minimize the total distance . The shortest path problem of a graph is to determine a sequence of edges, the total length of which is minimum. Oftentimes, these problems are computationally difficult–e.g., NP-complete or NP-hard–and, therefore, require extensive research.

### Simulation

Permutations and combinations can be employed for simulations in many areas. Permutations representing various genotype-phenotype associations are employed in genetics simulations (Doerge and Churchill, 1996). Other areas that employ permutations and combinations for simulations include networks, cryptography, databases and OR.

### Homeland security

This is a very specialized problem domain that has become a major national challenge after 9/11. To confront this challenge, many intelligent computing techniques have been applied, including intelligent pattern analyses of human faces, X-ray images, chemical components, data from a distributed network of wireless sensors, etc. Natural language processing and data mining techniques have been applied to sift through and monitor the tremendous

### Accumulation of electronic communication data.

Since combinatorics are extensively applied to these intelligent computing techniques, there is a wide spectrum of potentials for the national security issue. Some specific examples may include string searching algorithms and their performance analysis in communication data, pre- and post-analysis of combinatorial sequences of information elements, and combinatorial pattern matching.