2021 AMC 12B Fall Problem 23

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

Use linearity of expectation over the

29

possible consecutive pairs

(i, i+1)

The probability that both

i

and

i+1

are chosen is

\dfrac{5}{30} \cdot \dfrac{4}{29}

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.

Problem 23 in other years