# Combinatorics - History Of Combinatorics, Enumeration, Binomial Coefficients, Equivalence Relations, Recurrence Relations, Graph Theory

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objects study research computer

Combinatorics is the study of combining objects by various rules to create new arrangements of objects. The objects can be anything from points and numbers to apples and oranges. Combinatorics, like **algebra**, numerical analysis and **topology**, is a important branch of **mathematics**. Examples of combinatorial questions are whether we can make a certain arrangement, how many arrangements can be made, and what the best arrangement for a set of objects is.

Combinatorics has grown rapidly in the last two decades making critical contributions to computer science, operations research, finite **probability theory** and cryptology. Computers and computer networks operate with finite data structures and algorithms which makes them perfect for enumeration and graph theory applications. Leading edge research in areas like neural networking rely on the contribution made by combinatorics.

Combinatorics can be grouped into two categories. Enumeration, which is the study of counting and arranging objects, and graph theory, or the study of graphs.

## Additional Topics

Leonhard Euler (1701-1783) was a Swiss mathematician who spent most of his life in Russia. He was responsible for making a number of the initial contributions to combinatorics both in graph theory and enumeration. One of these contributions was a paper he published in 1736. The people of an old town in Prussia called Königsberg (now Kaliningrad in Russia) brought to Euler's attention…

To enumerate is to count. In combinatorics, it is the study of counting objects in different arrangements. The objects are counted and arranged by a set of rules called equivalence relations. One way to count a set of objects is to ask, "how many different ways can the objects be arranged?" Each change in the original arrangement is called a permutation. For example, changing the ord…

The importance of binomial coefficients comes from another question that arises. How many subsets are contained in a set of objects? A set is just a collection of objects like the three songs on the CD. Subsets are the set itself, the empty set, or the set of nothing, and any smaller groupings of the set. So the first two songs alone would be a subset of the set of three songs. Intuitively, eight …

Graphs are sets of objects which are studied based on their interconnectivity with each other. Graph theory began when people were seeking answers to questions about whether it was possible to travel from one point to another, or what the shortest distance between two points was. A graph is composed of two sets, one of points or vertices, and the other of edges. The set of edges represents the ver…

Trees are yet another type of graph. Trees have all the properties of graphs except they must be connected with no cycles. A computer's hard drive directory structure is set up as a tree, with subdirectories branching out from a single root directory. Typically trees have a vertex labeled as the root vertex from which every other vertex can be reached from a unique path along the edges. Not…

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