2023 AMC 12A Problem 21

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 \overline{AB} is an edge of the polyhedron, then d(A,B)=1, but if \overline{AC} and \overline{CB} are edges and \overline{AB} is not an edge, then d(A,B)=2. Let Q, R, and S be randomly chosen distinct vertices of a regular icosahedron (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.