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.)

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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.

Tagged: expected value · sampling without replacement