The overall goal of this paper is to give a method of computing out how many words of length n there are for any Coxeter group via its Brink-Howlett automaton. [6] [7] To build our automaton, we focus on Coxeter systems and root systems honing in on a special set of roots called the small roots. We follow closely [1] [5] for the first two chapters. Finally, we build the Brink-Howlett automaton through literature compiled through the years and present explicit examples of A˜1 and the Coxeter group on three generators which each pair of generators is in a free relation with one another.