2013 AMC 10B Problem 12

Below is the professionally curated solution for Problem 12 of the 2013 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2013 AMC 10B solutions, or check the answer key.

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Concepts:basic probability regular polygon sampling without replacement

Difficulty rating: 1220

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?

$\dfrac{2}5$

$\dfrac{4}9$

$\dfrac{1}2$

$\dfrac{5}9$

$\dfrac{4}5$

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 .

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