2025 AMC 12A Problem 13

Let C = \{1, 2, 3, \ldots, 13\}. Let N be the greatest integer such that there exists a subset of C with N elements that does not contain five consecutive integers. Suppose N integers are chosen at random from C without replacement. What is the probability that the chosen elements do not include five consecutive integers?

\textbf{(A)} \dfrac{3}{130} \qquad \textbf{(B)} \dfrac{3}{143} \qquad \textbf{(C)} \dfrac{5}{143} \qquad \textbf{(D)} \dfrac{1}{26} \qquad \textbf{(E)} \dfrac{5}{78}

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