2013 AMC 10B Problem 12

Question

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?

\dfrac{2}5

\dfrac{4}9

\dfrac{1}2

\dfrac{5}9

\dfrac{4}5

A regular pentagon has five equal sides and five equal diagonals

After one segment is chosen, count same-length choices among the remaining nine

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 .

Problem 12 in other years