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What is combinatorics?

What is combinatorics? Here are some definitions.

Noun
  1. (mathematics) a branch of mathematics that studies (usually finite) collections of objects that satisfy specified criteria
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Examples
The reason is a topic for another note, on the limitations of combinatorics in geometry.
A variety of applications to extremal combinatorics and computational complexity are discussed, in addition to a number of open problems.
His early works range over number theory, statistics, combinatorics, game theory, as well as his principal interest of commutative algebra.
His thesis work combined algebra and combinatorics into the new field of matrix theory.
His work led to considerable activity in the new area of algebraic combinatorics.
The problems which attracted him most were problems in combinatorics, graph theory, and number theory.

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