Enumeration: the counting of a set of objects - the set of odd numbers below 100, the set of classes I need to graduate, the set of trees in my backyard,  whatever.
Optimization: Given a set of objects that interact together according to a set of rules, what can to be done to optimize the interaction with some goal in mind? For example, how can I tell a salesman how to visit ten cities once each, travelling the shortest possible distance?
Existence: Given a certain set, constructed according to a set of rules, can the existence of an object in that set with specified properties be proved? For example, is there a way to color a map with four colors so that no two adjacent countries have the same color?
Algorithms: This really isn't a separate area, because investigating each of the other three usually produces algorithms for computing structures. 

Etymology: Combinatorics shares the same root with the word combinations, and indeed probably came into usage originally as the study of counting the number of combinations of performing a certain action or picking a hand of cards or some such. 

Many mathematicians believe that Combinatorics is not a legitimate field unto itself, but rather one has a "combinatorial approach" to another "real" field, like algebra, analysis, or topology. In my opinion, a method of thinking that has been so incredibly productive ought to have its own field within mathematics. Why, for example, a large portion of the underpinings of computer science is combinatorial. Plus, combinatorial approaches often have the benefit of producing both visually intuitive proofs, and algorithms for computation. 

        -- Robert Ellis, University of CA - San Diego Mathematics Graduate Student 

Last Updated October 29, 2007