AMC 12 2013a Problem 24

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular $12$-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?

\textbf{(A)} \ \frac{553}{715} \textbf{(B)} \ \frac{443}{572} \textbf{(C)} \ \frac{111}{143} \textbf{(D)} \ \frac{81}{104} \textbf{(E)} \ \frac{223}{286}