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How many five-card hands can be chosen from exactly 2 suits of an ordinary 52-card deck? There are 4 suits: clubs, diamonds, hearts, and spades.

I think it would be (26 C 5) but I not sure if I am interpreting the question correctly.

mathisfun1234
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    Well, that would be the number of ways to choose $5$ Black cards, or $5$ Red cards. But there are several ways to choose $2$ suits. And you have to worry about overlaps. – lulu May 07 '20 at 23:22
  • Can you explain it more? I am not understanding it. Would it be (26C5) * (4C2)? – mathisfun1234 May 07 '20 at 23:35
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    Almost, but there is still a problem. That formula counts, say, a spade flush multiple times. And, I'd say that the word "exactly" in the problem says that you should not count it at all. So, if you agree with my reading of the problem, you have to subtract off all the flushes you counted (multiple times). Even if you don't agree with my reading you still have to subtract the extra copies you counted. – lulu May 07 '20 at 23:41

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There are $\binom{4}{2}$ ways to choose the two suits. Then there are $\binom{13}{k}\binom{13}{5-k}$ ways to choose a hand such that $k$ are from the first suit and the rest are from the other suit. Thus $\binom{4}{2}\sum_{k=1}^4\binom{13}{k}\binom{13}{5-k}$ hands.

dromastyx
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