How many 5 card poker hands only have a high card, i.e no pair, straight, flush, e.t.c? A breakdown of high cards would also be appreciated.

I do know what is special about 23457 off-suited - any card improves it.

  • Not entirely sure what you mean? Any hand without at least a pair is a "high card" hand. Are you looking for a formula to calculate how many there are?
    – Toby Booth
    Jan 26, 2015 at 13:44
  • I can't see how you would go about it other than laying out every hand and counting those whose highest score is high card.
    – JMP
    Jan 26, 2015 at 14:51
  • See this link en.wikipedia.org/wiki/Poker_probability
    – paparazzo
    Apr 13, 2017 at 22:58

2 Answers 2


From the Wikipedia article on Poker Hands:

Of the 2,598,960 possible five-card combinations, {formula elided} 1,302,540 do not contain any pairs and are neither straights nor flushes. As such, the probability of being dealt "no pair" in a five-card deal is approximately 50.11%

In seven-card poker, the frequency of such "no pair" is 23,294,460; the probability of being dealt one is approximately 17.4%.[4][Note 3]

The article also has the formula for calculating it, which didn't translate well to this site.


If you mean distinguishable hands, 1277.


If you mean all hands, dantiberian's 1302540 is correct.

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