How deep is your playbook?

An American football season lasts amere sixteen games played over seventeenweeks. This compact season leads to, fairly or unfairly, intense scrutiny of every player’s performance and each coach’s decision.Our goal in this paper is to determine a measure of complexity for the decision of choosing a defensive alignment on any given down. There are only four standard defensive formations, defined generally by the personnel on the field, but how a coach physically situates the players on the field can emphasize widely different defensive strengths and weaknesses. To describe the number of ways a coach achieves this goal, we utilize the notion of equivalence classes from abstract algebra to define classifications of defensive formations. Enumerative combinatorics is then necessary to count the number of fundamentally different defensive alignments through the application of binomial coefficients.Descriptions of the rules for the game and diagrams of different defensive alignments make this paper accessible to even the novice, or non-fan.