Number of 5 card stud poker hands

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?

• Is $P$ a binomial coefficient or maybe something else? – dtldarek Mar 13 '12 at 9:08
• @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

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 $$P(52,5)=\binom{52}5=52\cdot51\cdot50\cdot49\cdot48\;.$$