2008 AMC 12B Problem 22

Problem 22 of 25HarderCounting & Probability

A parking lot has

16

spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose their spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires

2

adjacent spaces. What is the probability that she is able to park?

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Solution

After the

12

cars park,

4

spaces are empty, equally likely to be any

4

of the

16,

for

\binom{16}{4} = 1820

equally likely sets. Auntie Em fails exactly when no two empty spaces are adjacent. The number of ways to place

4

non-adjacent empties among

16

\binom{13}{4} = 715.

So the probability she can park is

1 - \frac{715}{1820} = \frac{1105}{1820} = \frac{17}{28}.

Thus, the correct answer is E.

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