Combinatorics

Summary

Combinatorics is the part of mathematics that studies finite or countable discrete structures. It asks three main types of question about such structures. Enumerative questions ask how many structures of a given type exist. Existence questions ask if a structure with given properties exists. Extremal questions ask how large or small a structure with given properties can be.

Its basic objects are finite sets, arrangements of elements, subsets, partitions and graphs. Its basic tools are counting rules, bijections, recurrences, generating functions and the pigeonhole principle.

Remarks

The rule of sum and the rule of product are the first counting rules. Binomial coefficients count the subsets of a given size of a finite set. Graph theory is a large subfield of combinatorics. Its basic notion is a graph.

Explanations

L1 human

Combinatorics is the mathematics of counting and arranging. A typical question is: in how many ways can 5 people sit in a row? The answer is $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$, and combinatorics gives general methods for questions of this type.