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In a game of euchre using a 24-card deck, where each player is dealt 5 cards, the number of possible hands can be calculated using combinations. Specifically, the number of ways to choose 5 cards from a 24-card deck is given by the combination formula ( \binom{n}{k} ), which is ( \binom{24}{5} = \frac{24!}{5!(24-5)!} = 42,504 ). Thus, there are 42,504 possible euchre hands.

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