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Probability of being divisible by 3

Source: AMC 12B 2010, Problem 16

February 25, 2010
probabilityAMC

Problem Statement

Positive integers a,b, a,b, and c c are randomly and independently selected with replacement from the set {1,2,3,…,2010}. \{ 1,2,3,\dots,2010 \}. What is the probability that abc \plus{} ab \plus{} a is divisible by 3 3? <spanclass=′latex−bold′>(A)</span> 13<spanclass=′latex−bold′>(B)</span> 2981<spanclass=′latex−bold′>(C)</span> 3181<spanclass=′latex−bold′>(D)</span> 1127<spanclass=′latex−bold′>(E)</span> 1327 <span class='latex-bold'>(A)</span>\ \dfrac{1}{3} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{29}{81} \qquad<span class='latex-bold'>(C)</span>\ \dfrac{31}{81} \qquad<span class='latex-bold'>(D)</span>\ \dfrac{11}{27} \qquad<span class='latex-bold'>(E)</span>\ \dfrac{13}{27}
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