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5 cards from a deck of 52 , how many ways of four of a kind can be dealt ?

I have (13c1) to determine the rank of card

so (13c1)(4c4) is all i can think of until now

the answer is 2(13c2)(4c4)(4c1)

can someone explain how does this work ?

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There are indeed $\binom{13}{1}$ ways of choosing the kind we have $4$ of. And once that is done, that part of the hand is determined. But then there is the useless fifth card, which can be chosen in $\binom{48}{1}$ ways.

So the number of $4$ of a kind hands is $(13)(48)$. For the probability, divide by $\binom{52}{5}$.

Remark: The person who did the counting you quote did it in I think less clear way. The $4$ of a kind hand will have two denominations, which can be chosen in $\binom{13}{2}$ ways.

For each choice, we have $2$ choices as to which denomination we will have $1$ of. And then the actual card can be chosen in $\binom{4}{1}$ ways, for a total of $2\binom{13}{2}\binom{4}{1}$.

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  • $\begingroup$ why does the answer give a 2(13c2)(4c4)(4c1) ? what is 2(13c2) representing ? $\endgroup$ – cindy Oct 17 '13 at 4:00
  • $\begingroup$ I added a remark at the end explaining how that is obtained. It gives the same number as my way. $\endgroup$ – André Nicolas Oct 17 '13 at 4:04
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Hint: it's easy to work out how many sets of 4 cards are a 4-of-a-kind.

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