Combinatorics studies problems involving finite sets of objects that are defined by certain specified properties. For example, the objects in question may themselves be sets, numbers, graphs or other geometrical configurations.
Enumerative combinatorics is concerned with counting the number of objects of a certain kind.
Extremal combinatorics is concerned with finding the optimal objects of a certain kind.
Topological methods, algebraic methods and even probabilistic methods have been used to solve combinatorial problems.
Computer algorithms have also been used to solve some seemingly intractable combinatorial problems.
Conversely, combinatorial methods have been used successfully to solve problems in many areas of mathematics and computer science.
Here is a sample problem that would use combinatorics:
Strangers and Acquaintances (F.P. Ramsey 1930):
What is the least number of people that you need to have in a room so that there is always a group of three mutual strangers or a group of three mutual acquaintances? The answer is six.

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A=B

Downloadable combinatorics text by Marko Petkovsek, Herbert Wilf and Doron Zeilberger. Published by A. K. Peters.

Algebraic Combinatorics via Finite Group Actions

A hypertext by A. Betten, H. Fripertinger and A. Kerber.

ArXiv Front: CO Combinatorics

Combinatorics section of the Front for the Mathematics ArXiv.

Combinatorial Data

Brendan McKay's collection of miscellaneous tables in graph6 or sparse6 format.

Combinatorics

MathWorld article with basic definitions and links.

The Combinatorics Net

Maintained by Bill Chen.

The Cutting Sticks Problem

An unsolved combinatorial problem by Frans Faase.

Encyclopedia of Combinatorial Structures

Searchable database of structures with specifications and counts.

Enumeration of Polyhedra

Enumeration of polyhedra with up to 9 faces by Steven Dutch.

Enumeration of the Binary Trees

A note on Catalan numbers by Ivan Galkin.

Extremal Combinatorics

With Applications in Computer Science by Stasys Jukna.

G-Systems

Theory of chord structures to create music through algorithms by Vladimir Ladma.

Generating Functions

Introduction to generating functions with interactive java applets by Alexander Bogomolny.

The Math Forum Math Library - Combinatorics

Comprehensive catalog of websites relating to Combinatorics.

Polya's Enumeration Theorem

An article on generating functions and Polya's theory of counting by Karl Dahlke.

Series Expansions

Iwan Jensen counts polyominoes (aka lattice animals), paths, and various related quantities.

Squared Squares

Tilings of squares and rectangles, the classical problem and variations.

Topics in Mathematics - Combinatorics

In the Mathematics Archive at University of Tennessee, Knoxville.

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