vpFREE FAQ_Probability

Probability Computations

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.