2005 AMC 10A Problem 9

Below is the professionally curated solution for Problem 9 of the 2005 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 10A solutions, or check the answer key.

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Concepts:basic probability combinations

Difficulty rating: 1280

Three tiles are marked X and two other tiles are marked O. The five tiles are randomly arranged in a row. What is the probability that the arrangement reads XOXOX?

$\dfrac{1}{12}$

$\dfrac{1}{10}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

Solution:

The three X positions can be any of $\binom{5}{3} = 10$ equally likely choices, and exactly one of them produces XOXOX. So the probability is $\dfrac{1}{10}.$

Thus, the correct answer is B.

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