2013 AMC 12A Problem 20

Let S be the set \{1,2,3,...,19\}. For a,b \in S, define a \succ b to mean that either 0 < a - b \le 9 or b - a > 9. How many ordered triples (x,y,z) of elements of S have the property that x \succ y, y \succ z, and z \succ x?

\textbf{(A)} \ 810 \qquad \textbf{(B)} \ 855 \qquad \textbf{(C)} \ 900 \qquad \textbf{(D)} \ 950 \qquad \textbf{(E)} \ 988

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