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May 9th, 2016, 10:44 PM   #1
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a card game in probability

4 cards are dealt from a 52-card deck. How many hands contain 2 distinct pairs?

This is how I do it.

From the 13 ranks, I choose 1.
In this rank, I choose 2 cards from 4.

From the remaining 12 ranks, I choose 1.
In this rank, I choose 2 cards from 4.

Now, I do the following calculations.

13C1*4C2*12C1*4C2 = 5616.

But the answer key says 2808.

If I divide my answer by 2, I will get the right answer. But I don't why I have to do that. Please explain it.

Thank you very much.
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May 10th, 2016, 12:51 AM   #2
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You double-count each pair because, for example, A + 5 and 5 + A are both counted.
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May 10th, 2016, 10:57 AM   #3
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In my calculation, I choose 2 cards from 4 in one rank and do the same thing in a different rank. I don't see how double-counting occurs.
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May 11th, 2016, 09:15 PM   #4
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I have thought about my solution for some time. Because I am using combinations where the order does not matter, arrangements of the same objects count only once. For example, abc acb bac bca cab cba. There is only one combination of the three letters. I am confused about why I have to divide my answer by two.
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