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Given a 52 card deck how many 5 card stud poker hands are there?

5 card stud poker is when 1 card is dealt face down and 4 face up. I guessed it would be $\frac{P(52,5)}{4!}$ because the order of the face up cards doesn't matter, but Schaum's says it's just $P(52,5)$.

Why?

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Is $P$ a binomial coefficient or maybe something else? –  dtldarek Mar 13 '12 at 9:08
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@dtldarek: $P(n,k)=\binom{n}k k!$. –  Brian M. Scott Mar 13 '12 at 9:13
    
@BrianM.Scott Thanks. I am always confused by those. It keeps me wondering, why won't we create yet another symbol for $\binom{n}{k}k!(n-k)!$. –  dtldarek Mar 13 '12 at 10:36

1 Answer 1

up vote 1 down vote accepted

Revised: The order in which the cards are dealt matters, because there is a round of betting after each up card appears. Thus, there is a difference between getting the ace of spades as a hole card followed in order by $\diamondsuit Q,\diamondsuit 10,\diamondsuit 3,\clubsuit Q$ and the same hole card followed in order by $\diamondsuit Q,\clubsuit Q,\diamondsuit 10,\diamondsuit 3$: the players are likely to be quite differently. Thus, the correct answer really is $$P(52,5)=\binom{52}5=52\cdot51\cdot50\cdot49\cdot48\;.$$

Your answer would be correct if one could ignore the order in which the cards are dealt and distinguish hands only according to which cards they contain and which one is hidden.

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"However, your answer is correct if one ignores order and distinguishes hands only according to which cards they contain and which one is hidden." What you mean is that the hand "2 up 3,4,5,6 down" is the same as "2 up, 6,5,4,3 down"? That's how I would interpret it - in fact I don't see any other valid interpretation. –  Robert S. Barnes Mar 13 '12 at 9:35
    
@Robert: (You have up and down reversed.) No, the order really does matter, though I had to remember just how the game is played to realize it: see my revised answer. –  Brian M. Scott Mar 13 '12 at 9:44
    
Crappy question. They don't mention anything in the book about the betting occurring between each round of dealing an up card. I agree with you that in that case order of the 4 up cards matters, but without that piece of information I would argue that my answer is the correct one. Thanks for the added info and revision! –  Robert S. Barnes Mar 13 '12 at 10:25

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