2) How many different 13-card bridge hands contain a singleton in hearts and two five-card suits? [A5 marks](A singleton is exactly one card of a particular suit. A five-card suit is exactly five cards of the same suit.)
Question
- How many different 13-card bridge hands contain a singleton in hearts and two five-card suits? [A5 marks](A singleton is exactly one card of a particular suit. A five-card suit is exactly five cards of the same suit.)
Solution
To solve this problem, we need to break it down into several steps:
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First, we need to choose which suit will be the singleton. Since we know it's hearts, this step is already done for us.
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Next, we need to choose which card in the hearts suit will be the singleton. There are 13 cards in a suit, so there are 13 choices for this step.
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Then, we need to choose which two suits will be the five-card suits. There are 3 remaining suits (spades, diamonds, clubs), and we need to choose 2 of them. This can be done in 3 ways.
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For each of the two five-card suits, we need to choose which 5 cards will be in the suit. There are 13 cards in a suit, and we need to choose 5 of them. This can be done in "13 choose 5" ways, or 1287 ways. Since we need to do this twice (once for each suit), the total number of ways is 1287 * 1287 = 1,656,769.
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Finally, we need to choose the remaining 2 cards from the remaining suit. There are 13 cards left in this suit, and we need to choose 2 of them. This can be done in "13 choose 2" ways, or 78 ways.
Multiplying all these together, we get the total number of different 13-card bridge hands that meet the criteria:
13 (singleton choices) * 3 (suit choices) * 1,656,769 (five-card suit choices) * 78 (remaining card choices) = 390,564,894,834 different hands.
Similar Questions
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