But some of the cards you can pick for the 13th card match a pair, so it would be easy to double-count.Ħ pairs, no trips: (13C6)* 6^6 * 28 cards left over = 2,241,727,488 possibilitiesġ quad + 4 pairs, no trips: 13 * (12C4) * 6^4 * 32 cards = 266,872,320Ģ quads + 2 pairs, no trips: (13C2) * (11C2) * 6^2 * 36 cards = 5,559,840 The brute force approach: 6 pairs can be made up of either distinct pairs or quads. How did you calculate 6 pairs? I'm not able to come up with an easy way. Some of these will be dragons in addition to being 3-flushes, but obviously only a small fraction. So, we have 5,705,516,392 hands that can be played as 3-flushes, giving odds of about 110:1.
Percentage % of a A-K (including A-K suited): 0.01057%Ĭounting or not counting the 4 A-K suited hands doesn't make any significant difference.ģ-flushes are a bit harder because some of them could be two-flushes, although the effect of those cases is pretty small.
# of 13-card hands consisting of A-K with any suits = number of ways to pick one suit for each of 13 cards = 4^13 = 67,108,864
