Combinatorics

Combinatorics is a branch of mathematics that concerns itself, at the elementary level, with counting things. For example, suppose that you have four dresses, but that you only have room for two in your suitcase, in how many ways can you choose these two dresses? The answer is six: if, for example, the dresses are green, red, pink and black, then the combinations you can choose are: green + red, green + pink, green + black, red + pink, red + blank, and pink + black. More generally, the number of ways you can choose k objects out of n is a binomial coefficient.

This problem is part of enumerative combinatorics, the part that focuses on enumerating and counting combinations of objects satisfying certain properties. Tools in enumerative combinatorics include generating functions and the umbral calculus. Combinatorics also studies codes, designs, finite geometries and Latin squares. Other branches are algebraic combinatorics and extremal combinatorics. Graph theory is sometimes also considered a part of combinatorics.