Combinatorics is a branch of mathematics that focuses on the permutation, enumeration, and combination of sets of elements. Sometimes, mathematicians use the word “combinatorics” to denote a larger subset of discrete math that includes graph theory. What is commonly referred to as combinatorics, in that case, is then called “enumeration’.

First combinatorial problems were studied by ancient Greek, Arabian and Indian mathematicians. Interest in combinatorics increased during the 19th and 20th century, with the development of graph theory and the four color theorem. Blaise Pascal, Jacob Bernoulli and Leonhard Euler are some of the leading mathematicians.

The most classical area of combinatorics is enumerative combinatorics. It focuses on counting the number of some combinatorial objects. While counting the number of components in a set is a quite broad mathematical problem, numerous problems that arise in applications boast of a relatively simple combinatorial description.

Analytic combinatorics focuses on the combinatorial structures enumeration using tools from probability theory and complex analysis. Compared to enumerative combinatorics that use clear generating functions and combinatorial formulae to describe the outcomes, the aim of analytic combinatorics is to obtain asymptotic formulae.

Other subfields and approaches of combinatorics are partition theory, design theory, finite geometry, order theory, matroid theory, extremal combinatorics, probabilistic combinatorics, and algebraic combinatorics, combinatorics on words, infinitary combinatorics, arithmetic combinatorics and topological combinatorics.

Combinatorics has numerous applications in other mathematics areas, including coding and cryptography, graph theory, and probability.

References

https://mathigon.org/world/Combinatorics

http://mathworld.wolfram.com/Combinatorics.html

https://www.dartmouth.edu/~chance/teaching_aids/books_articles/…/Chapter3.pdf