  Given a collection of points in the plane, classifying which subsets are
collinear is a natural problem and is related to classical geometric
constructions. We consider collections of points in a projective plane over a
finite field such that no three are collinear. This is a finite set and its
size is both combinatorially interesting and has deeper topological
consequences. We count the number of such collections classified by the
algebraic symmetries of the finite field. Variations of this problem have been
considered by Glynn, Bergvall, Das, O'Connor et al. We obtain the counts for 7
points over fields of characteristic 2. These new counts are governed by the
existence and classification of a configuration of points called the Fano
plane.
