2013 AMC 10B Problem 12

Let

S

be the set of sides and diagonals of a regular pentagon. A pair of elements of

S

are selected at random without replacement. What is the probability that the two chosen segments have the same length?

Show solution

Solution

A regular pentagon has

5

sides of one length and

5

diagonals of another length. After the first segment is chosen, there are

9

segments left, and exactly

4

of them have the same length as the first chosen segment. Therefore the probability is

\frac49

. Thus, the correct answer is B .