Odds of no pairs

A "no pair hand" can be defined as a hand that doesn't fit into any other category. The number of no pair hands can be found by subtracting the number of hands in the other categories from the total number of hands. If you do this you get the following result:

n(no pair hands) = n(total hands) - n(hands that beat ace-high) = (2 598 960 - 1 296 420) no pair hands = 1 302 540 no pair hands


NB: You could also calculate the number of no pair hands by calculating kicker combos where all five cards have to have different ranks, and subtract the number of straights and flushes from this. 4 out of the 1024 possible suit combinations for hands with five cards of different ranks will be flushes.