2023 AMC 10A Problem 25

If A and B are vertices of a polyhedron, define the distance d(A, B) to be the minimum number of edges of the polyhedron one must traverse in order to connect A and B. For example, if AB is an edge of the polyhedron, then d(A, B) = 1, but if AC and CB are edges and AB is not an edge, then d(A, B) = 2. Let Q, R, and S be randomly chosen distinct vertices of a regular icosahedron (a regular polyhedron made up of 20 equilateral triangles). What is the probability that d(Q, R) \gt d(R, S)?

\textbf{(A)} \dfrac{7}{22} \qquad \textbf{(B)} \dfrac{1}{3} \qquad \textbf{(C)} \dfrac{3}{8} \qquad \textbf{(D)} \dfrac{5}{12} \qquad \textbf{(E)} \dfrac{1}{2}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.