Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.
-- Ernst Mach
I suspect that Herr Mach was exaggerating a little - at least by modern standards. Although in many ways he wasn't too far off the mark. The Fundamental Theorem of Arithmetic talks about the prime factorization of numbers which is a fancy way of saying how many ways one number can be split up into smaller ones. The Fundamental Theorem of Algebra talks about how many roots a polynomial equation has. The Fundamental Theorem of Calculus tells us how much area is underneath a function between two points. (Okay - calling area a counting problem is a bit of a stretch but hopefully you get my point.)
In this chapter, we're going to talk about the ways mathematicians have developed for counting things and some of the conclusions that can be drawn from those methods.
Level of Difficulty
This chapter uses no mathematical techniques beyond arithmetic. That may sound pretty simple for a course at this level but wait until you see what we do with it. This chapter is a great demonstration of the way combinatorics takes some very simple techniques and applies them in creative ways to answer some relatively complex questions.