Here is the formula for calculating unique combinations:
N=number of possible values (e.g. 52 for a 52-card deck, 47 for the redraw)
P=number of positions (e.g. 5 for 5-card initial deal)
Simplest form:
(N!)
C = ------------
(P!)(N-P)!
where N! means 'N factorial', or 'N times N-1 times N-2 etc etc', all the way down to 1. (N-P)! in the numerator and denominator cancel each other out, so this reduces to:
N x (N-1) x ... x (N-P+1)
C = ---------------------------------
P x (P-1) x ... x 2
[ x 1 ]
where C is the number of unique combinations.
Example - the number of unique deals (i.e. combinations) from a 53-card deck is computed as follows:
N=53 (53-card deck)
P=5 (5 positions i.e. 5 card deal)
(53!)
C = ----------
(48!)(5!)
53 x 52 x 51 x 50 x 49
C = ----------------------------
5 x 4 x 3 x 2
[ x 1 ]
C = 2,869,685
Other similar questions can be answered like this, or based on this result. For example, there are 4 possible royals (one of each suit), so this occurs every 2,869,684 / 4 = 717,421.25 hands.
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