2000 AIME II Problems/Problem 3
A deck of forty cards consists of four 's, four 's,..., and four 's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let be the probability that two randomly selected cards also form a pair, where and are relatively prime positive integers. Find
There are ways we can draw two cards from the reduced deck. The two cards will form a pair if both are one of the nine numbers that were not removed, which can happen in ways, or if the two cards are the remaining two cards of the number that was removed, which can happen in way. Thus, the answer is , and .
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