This area is one of the oldest parts of combinatorics, such as in Kirkman's schoolgirl problem proposed in 1850.

The solution of the problem is a special case of a Steiner system, which systems play an important role in the classification of finite simple groups.

In the Ostomachion, Archimedes (3rd century BCE) considers a tiling puzzle.

Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. Graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem.

In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject.

Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects.

Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description.

Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting.

Combinatorics is well known for the breadth of the problems it tackles.

Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

A mathematician who studies combinatorics is called a Basic combinatorial concepts and enumerative results appeared throughout the ancient world.

To fully understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon.

Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained." One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella.

Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.