2021 Fall AMC 12B Problem 23

What is the average number of pairs of consecutive integers in a randomly selected subset of 5 distinct integers chosen from the set \{ 1, 2, 3, \ldots , 30\}? (For example the set \{1, 17, 18, 19, 30\} has 2 pairs of consecutive integers.)

\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{29}{30} \qquad\textbf{(E)}\ 1

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