TITLE: What is combinatorics?
QUESTION [18 upvotes]: I've tried to search the web and in books, but I haven't found a good definition or definitive explanation of what combinatorics is.

Could anyone give me a definition/explanation of combinatorics, of what combinatorics is, and what it deals with?
References that contain an answer to the question are appreciated.

Edit: I sees that many are saying that combinatorics deal with counting, but that doesn't seem to be the correct answer, for two reasons: first of all saying that combinatorics is just about counting means, at least to me, putting it inside set theory, because it's there where you define and deal with the more wide concept of counting; another reason is that there are some branch of mathematics which usually fall under combinatorics but doesn't directly deal with counting: for instance combinatorial design doesn't explicitly deal with counting.

REPLY [3 votes]: According to Wikipedia:

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).