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What is the probability that a hand of 5 cards chosen randomly and without replacement from a standard deck of 52 cards (which has 4 kings and 4 queens) contains the king of spades, exactly 1 other king, and exactly 2 queens

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Answer: Approximately 0.0003047

As a fraction, it is equal to exactly 792/(2,598,960)

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Step-by-step explanation:

The king of spades must be in the hand, and we can pick any other king. There are 3 ways to pick the other king. Then there are 4 C 2 = 6 ways to pick the two queens. The 4C2 refers to the nCr notation. Or you could use Pascal's Triangle.

Overall, there are 3*6 = 18 ways to pick the king of spades, another king, and exactly 2 queens.

Then we have one more card to pick that isn't a king nor a queen. There are 4+4 = 8 cards that are either a king or queen, leaving 52-8 = 44 cards that are neither.

So there are 18*44 = 792 ways to pick a king of spades, another king, exactly 2 queens, and some other card that isn't a king nor queen.

This is out of 52C5 = 2,598,960 possible five card hands.

The probability we're after is roughly 792/(2,598,960) = 0.0003047

Side note: order does not matter with card hands.

User Alma
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