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Quintuplet of integers

Source: 12B #23

November 17, 2021
AMCAMC 12AMC 12 B

Problem Statement

What is the average number of pairs of consecutive integers in a randomly selected subset of 55 distinct integers chosen from the set {1,2,3,…,30}\{ 1, 2, 3, …, 30\}? (For example the set {1,17,18,19,30}\{1, 17, 18, 19, 30\} has 22 pairs of consecutive integers.)
<spanclass=′latex−bold′>(A)</span> 23<spanclass=′latex−bold′>(B)</span> 2936<spanclass=′latex−bold′>(C)</span> 56<spanclass=′latex−bold′>(D)</span> 2930<spanclass=′latex−bold′>(E)</span> 1<span class='latex-bold'>(A)</span>\ \frac{2}{3} \qquad<span class='latex-bold'>(B)</span>\ \frac{29}{36} \qquad<span class='latex-bold'>(C)</span>\ \frac{5}{6} \qquad<span class='latex-bold'>(D)</span>\ \frac{29}{30} \qquad<span class='latex-bold'>(E)</span>\ 1
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