Counting techniques are used predominantly in combinatorial fields, but are occasionally borrowed for a proofs in other branches of mathematics. This (unfinished) article is a quick resource to point the user towards some of the combinatorial techniques developed to count discrete objects, and specific examples of how counting techniques have been applied in different fields.
Discussion of Prerequisites
Certain counting techniques require almost no background knowledge. Basic combinatorial arguments, those that deal with finite arrangements of a finite number of objects, require a background in Algebra, and an understanding of the factorial function. More advanced combinatorics requires linear algebra and exposure to recurrence relations. The more advanced combinatorial techniques require a background in calculus and differential equations, as typical advanced techniques involve generating functions.
Combinatorics is the basis for counting techniques. If you are looking to learn how to solve certain problems, see the Combinatorics front page. Otherwise the techniques below should help to serve your purposes.