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Combinatorics is a branch of pure mathematics that deals with discrete (and typically finite) things. It has connections to many other branches of mathematics, including algebra, probability theory, ergodic theory, and geometry, as well as computer science and statistical physics. Combinatorics includes "counting" the objects that satisfy certain criteria (enumerative combinatorics), determining when the criteria can be met, and constructing and analysing objects that meet the criteria (as in combinatorial designs and matroid theory), finding the "largest," "smallest," or "optimal" objects (extremal combinatorics and combinatorial optimization), and discovering algebraic structures these objects may have (extremal combinatorics and combinatorial optimization), and finding the " (algebraic combinatorics). Though it has acquired sophisticated theoretical approaches, particularly since the late twentieth century, combinatorics is as much about problem solving as it is about theory building. Graph theory is one of the oldest and most accessible branches of combinatorics, with several natural linkages to other fields. The structure of combinatorial sets is the subject of numerous combinatorial patterns and theorems. These frequently concentrate on a set's partition or ordered partition.

The following is an example of a simple combinatorial question: 

How many different ways can a deck of 52 different playing cards be arranged? The answer is 52! (52 factorial), which is equal to about 8.0658. 10⁶⁷. Another example is: Is it possible to assign a certain number n of people to sets such that each person is in at least one set, each pair of people is in exactly one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person, given a certain number n of people? The answer is determined by n.

In computer science, combinatorics is widely used to estimate the number of elements in various sets. A combinatorialist or combinatorics is a mathematician who specialises in combinatorics.

Enumerative combinatorics:

Enumerative combinatorics is concerned with determining the number of different combinatorial items. It is the most well-known area of combinatorics, and Counting the number of items in a set, on the other hand, is a vast mathematical topic, and several practical problems have a rather easy combinatorial explanation. The Fibonacci number is the most basic example of an enumerative combinatorics issue. A uniform foundation for counting partitions, permutations, and combinations is provided by the twelvefold method.

Analytic Combinatorics:

Analytic combinatorics use tools from probability theory and complex analysis to enumerate combinatorial structures. In contrast to enumerative combinatorics, which utilises explicit combinatorial formulae and generating functions to express the answers, analytic combinatorics aims to find the asymptotic formula.

Extremal Combinatorics:

Extremal combinatorics is the study of set systems' extreme issues. In this case, the queries concern the largest feasible graph that satisfies certain properties. The largest triangle-free graph on 2n vertices, for example, is a complete bipartite graph Kn, n. It can be difficult to get the extremal answer of (n) precisely at times, and one can only make an asymptotic estimate.

Probabilistic Combinatorics:

The questions in probabilistic combinatorics are something like this: what is the probability of a given property for an arbitrary discrete object like a random graph? What is the average number of triangles in an arbitrary graph, for example? The chance of arbitrarily selecting an item having those attributes that is greater than 0 is also used to determine the existence of combinatorial objects with particular required properties using probabilistic techniques. In applications to extremal combinatorics and graph theory, this strategy, dubbed the probabilistic technique, proved to be extremely effective.

Combinatorics is an ever-expanding branch of mathematics. As computer technology collects more data of various types, more analysis, organisation, enumeration, and other tasks become necessary. Here's a list of areas where combinatorics has already been used:

Networks of communication

Cryptography

Network Safety

Computational Molecular Biology (CMB) is a branch of molecular biology

Architecture of Computers

Scientific Research

Pattern Recognition in Language

Simulation\sDatabases

Exploration of data

Research on Homeland Security Operations

Equations of Geometry

Graph Theory and Application in 3D Computational Design

In theory, combinatorial approaches could be useful for any subject that accumulates data and requires data tracking, organisation, and application. Combinatorial algorithms are used in nearly every graphical feature on personal computers. Collision detection is a simple example that is employed in modern video games. Two sets of elements are compared against each other for intersections in collision detection. The purpose is to determine if the items have collided or not. Any repeated vector graph position will be considered a collision, according to the constraint. This implies that any calculated points that do not equal a collision are ignored, while the values that do equal a collision are used to determine what the object does next, according to the game's programming. The application of various combinatorial methods differs depending on the researcher's or observer's goal. This means that the methods can be used to solve an infinite number of mathematical and real-world problems that affect people's daily lives. It might be as simple as making your smartphone's banking app or communications more secure.

The following are the subsets of combinatorics mathematics that are currently known:

Enumerative combinatorics: 

This is the most common type of combinatorics, and it is well known for the Fibonacci sequence.

Analytic Combinatorics:

Enumerative combinatorics uses tools from complex analysis and probability theory in analytic combinatorics.

Partition Theory:

Partition theory is a branch of number theory and analysis that has evolved into its own discipline in combinatorics.

Graph Theory:  

In combinatorics, graphs are essential. On graphs, the application of possibility and range is frequently represented.

Design Theory:

The study of combinatorial designs that are portions of subsets with intersecting attributes is known as design theory.

Finite Geometry:

The study of geometric systems with a finite structure is known as finite geometry.

Order Theory:

Order Theory is the study of finite and infinite partially ordered sets

Matroid Theory:

Matroid Theory is a branch of mathematics that studies the properties of sets of vectors in a vector space that are independent of coefficients in a linear dependency.

Extremal Combinatorics:

Extremal combinatorics is the study of set systems' extreme issues.

Probabilistic Combinatorics:

Probabilistic combinatorics is the study of the chances that a random discrete item will have a specific property.

Algebraic Combinatorics:

Algebraic combinatorics is the application of combinatorial approaches to algebra issues using group theory and representation theory, or other abstract algebra methods.

Geometric Combinatorics:

Combinatorics applied to convex and discrete geometry is known as geometric combinatorics.

Topological Combinatorics:

Combinatorial formulas are frequently employed to aid in topographical problems and topology research.

Arithmetic Combinatorics:

Combinatorial procedures that simply use addition and subtraction are known as arithmetic combinatorics.

Infinitary Combinatorics:

Combinatorial set theory, often known as infinitary combinatorics, is the implementation of combinatorics methods to infinite sets.

Combinatorics on Words:

Combinatorics on words is the application of combinatorics to the study of patterns in written language.

Ancient history contains basic principles about combination and enumerative conclusions. Many people have been credited with developing parts of what is now known as combinatorics, but according to MIT, Gian-Carlo Rota is the founder of contemporary enumerative combinatorics. They credit him with turning the methods from a collection of gimmicks to a serious subject. With his research in number theory and graph theory, Paul Erdos is frequently credited with creating much of modern combinatorics. Richard P. Stanley is the most recent exponent of enumerative combinatorics. In an academic setting, Stanley's two-volume work is used as the classic introduction to enumerative combinatorics mathematics. Combinatorics was not recognised to be its own field of study before Stanley's publication, and it was intertwined with many other subdivisions of mathematics. As a result, pinpointing the beginning of combinatorics as a whole is difficult, as thousands of mathematicians have contributed to and expanded on what it is.

Combinatorics can be found in almost every discipline, as one would anticipate from such a large subset of mathematics. Here are a few instances to assist illustrate where these techniques are beneficial:

Military Strategy:

In dangerous situations, resource and personnel management are vital to mission accomplishment and human survival. While the military favours credible intelligence from reliable sources, strategists must nevertheless analyse locations and options. Strategists can find every potential arrangement and narrow down the best possibilities using the information they can obtain, such as unit size, topology, resource availability, equipment, and transportation.

Visual Representation:

Literal graphs are the foundation of computer graphics. The computer maintains track of where things are and generates graphics based on a large graph that shows where things are and how they've changed. Graphs are used to depict the image exhibited even in 3D representations. 3D modelling software and the computer's operating system both use combinatorics formulas.

Engineering:

Material studies such as heat, wind, and rain are graphed using pattern and picture analysis. This data can then be utilised to assist shape sections that operate best when exposed to certain factors. It also aids prototyping by allowing a rapid study of limitation requirements to be performed.

Biomedicine:

Years of research are required in molecular biology before any testing can be implemented. This necessitates a thorough examination and mapping out of theoretical and analytical options.

Scheduling Transportation:

The arrangement of train, bus, and aeroplane timetables is a basic but important application of combinatorics. It's tough to match flights or trips to passenger demand while retaining the most efficient routes, but permutation calculations can help.

Design and Organization:

Whether you're organising an event or a coding project, combinatorial theory can assist you come up with the greatest design possible within your constraints.