Problem:
Two counterfeit coins of equal weight are mixed with 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 coins. A second pair is selected at random without replacement from the remaining coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all selected coins are genuine?
Answer Choices:
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Solution:
The weights of the two pairs of coins are equal if each pair contains the same number of counterfeit coins. Therefore either the first pair and the second pair both contain only genuine coins, or the first pair and the second pair both contain one counterfeit coin. The number of ways to choose the coins in the first case is . The number of ways to choose the coins in the second case is . Therefore the requested probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions