AMC 12 2025a Problem 13

Let $C = \{1, 2, 3, \dots, 13\}$. 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)}\frac{3}{130} \textbf{(B)}\frac{3}{143} \textbf{(C)}\frac{5}{143} \textbf{(D)}\frac{1}{26} \textbf{(E)}~\frac{5}{78}