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Probability of selecting 4 genuine coins

Source: 2011 AMC A Problem 21

June 25, 2011
probabilityconditional probabilityAMC

Problem Statement

Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?
<spanclass=′latex−bold′>(A)</span> 711<spanclass=′latex−bold′>(B)</span> 913<spanclass=′latex−bold′>(C)</span> 1115<spanclass=′latex−bold′>(D)</span> 1519<spanclass=′latex−bold′>(E)</span> 1516 <span class='latex-bold'>(A)</span>\ \frac{7}{11}\qquad<span class='latex-bold'>(B)</span>\ \frac{9}{13}\qquad<span class='latex-bold'>(C)</span>\ \frac{11}{15}\qquad<span class='latex-bold'>(D)</span>\ \frac{15}{19}\qquad<span class='latex-bold'>(E)</span>\ \frac{15}{16}
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