Combination: The game of euchre uses only 9s, 10s, jacks, queens, kinds, and aces...

wolfyy

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Oct 1, 2017
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The full question is:
The game of euchre uses only 9s, 10s, jacks, queens, kinds, and aces from a standard deck of cards. How many five-card hands have:
(a) at least two red cards?

My attempt:
12C2 x 22C3 = 101640

12C2 represent the two red card
22C3 represent all the card leftover after removing the two red cards. And since the restriction is met it wouldn't matter if I mix the red and black card together... right?

I understand that this question can be solved by:
All combination(24C5) - no red(12C5) - 1 red(12C1x12C4), and the answer should be 35772

At the same time, I don't understand what is wrong with my original logic, please help :D
 

gregk

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Sep 18, 2017
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My attempt:
12C2 x 22C3 = 101640
You're counting some hands more than once. Consider the hand formed by first selecting {9h, 10h}, and then selecting {Jh, 9s, 10s}. Now instead consider the hand formed by selecting {10h, Jh} and then selecting {9h, 9s, 10s}. You count these hands separately, but they are actually the same.

What you could do is count the number of hands having exactly 2 red cards, plus the number of hands having exactly 3 red cards, and so on.
 

wolfyy

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Oct 1, 2017
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Ah I see! Your example clears up my mind a bit. So I will have to use cases after all.

Thanks for the explanation, and I hope I wouldn't make this kind of mistake again:p
 
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