MathNet 2016-part-04 — 184 problems (EN–VI) Question 68

Let $K$ be a finite field of cardinality $q \ge 3$, and let $S$ be the set of degree $q-2$ polynomials in $K[X]$ whose coefficients are all non-zero and pairwise distinct. Determine the number of polynomials in $S$ having $q-2$ distinct roots in $K$.
Marian Andronache