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I am a little bit confused about this problem. It said that three cards are dealt from a standard deck of cards. In how many ways can one get at least one king?

Now If I use the complement, then I want the number of ways one can deal 3 cards minus the number of ways of not getting a king. Well, 52*51*50 = 132600 is the number of ways of dealing 3 cards. The number of ways not dealing king is 48*47*46 = 103776. So I'd 132600 -103776 = 28824 ways of getting at least one king. Is this reasoning right?

On the other hand if I do $\binom{4}{1}\binom{48}{2} + \binom{4}{2}\binom{48}{1} + \binom{4}{3}\binom{48}{0}$, I get a different answer. Am I doing something wrong?

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    The key phrase is "in how many ways." Given that "dealing" three cards usually involves the sequential giving of cards, the phrase "in how many ways" to me indicates that order matters. But the question is poorly phrased, making the author's intent far from obvious.2012-05-31

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The second result ist correct. The first one is not, since you're counting some configurations more than once.

I'd say the number of ways you can deal 3 cards is $\frac{52*51*50}{3*2*1}=22100$, the number of ways you can deal 3 cards with no king is $\frac{48*47*46}{3*2*1}=17296$. So the difference is $4804$, which agrees with the second result