Question

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.)

$\dfrac{2}{3}$

$\dfrac{29}{36}$

$\dfrac{5}{6}$

$\dfrac{29}{30}$

$1$

Answer: A

Concepts:expected value sampling without replacement

Difficulty rating: 2190

Solution:

For each of the $29$ adjacent pairs $(i, i+1),$ let an indicator be $1$ if both are in the subset. The probability of this is $\dfrac{5}{30} \cdot \dfrac{4}{29} = \dfrac{2}{87}.$

By linearity of expectation, the expected number of consecutive pairs is $29 \cdot \dfrac{2}{87} = \dfrac{2}{3}.$

Thus, the correct answer is A.

2021 AMC 12B Fall Problem 23

Answer: A

Difficulty rating: 2190

Solution:

Problem 23 in Other Years