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H)
how many five- card hands are possible that contain at least three kings?

User Tgallei
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Answer: 4560

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Step-by-step explanation:

We'll break things up into cases. The phrasing "at least 3" means "3 or more".

We either will have 3 kings or 4 kings.

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Case A) There are exactly 3 kings.

Use the nCr combination formula. There are n = 4 kings total and we select r = 3 of them. That gives 4C3 = 4 ways to pick the 3 kings.

Put another way: there are 4 ways to leave a certain king out of the hand.

Then we have 52-4 = 48 cards that aren't a king, and we need to fill the remaining r = 2 slots. Through the nCr formula, you should find that 48C2 = 1128

To recap, we found

  • 4 ways to pick the three kings
  • 1128 ways to pick the other two cards that aren't kings

That gives 4*1128 = 4512 ways to have case A happen.

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Case B) There are exactly 4 kings

4C4 = 1, so there's only one way to select all the kings. The order doesn't matter. Then we have 48 cards to pick from for the fifth slot.

Overall there are 1*48 = 48 ways to have case B play out.

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There were 4512 ways to have case A happen, and 48 ways to have case B happen.

These cases are mutually exclusive to allow us to simply add the counts:

4512+48 = 4560

This is why the final answer is 4560

User Simonauner
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