1. Total Number Of Poker Hands
2. Number Of Two Pair Poker Hands
3. Number Of Five Card Poker Hands

There are two ways to derive the number of high-card hands. One is by realizing that the set of high-card hands is the complement to the set of all other hands. That means the number of high card hands is 2598960 – 40 – 624 – 3744 – 5108 – 10200 – 54912 – 123552 – 1098240 = 1,302,540. Though the probability of any one hand being dealt to a player doesn't diminish with number of players, the strength of a hand should increase as fewer players are available. For example: a J7 hand would be pretty crap on a table with 10 players as there's a high probability that at least 1 other player has an equal or higher hand (it's an. Official Poker Hand Rankings Know your poker hand order. A poker hand consists of five cards, which fall into several categories. Below is the complete list of poker hands, from highest to lowest. Please note that hand rankings for Short Deck are slightly different. Find out more about the short deck poker hand rankings on the dedicated page.

Brian Alspach

13 January 2000

Abstract:

The types of 3-card poker hands are

• straight flush
• 3-of-a-kind
• straight
• flush
• a pair
• high card

The total number of 3-card poker hands is .

A straight flush is completely determined once the smallest card in thestraight flush is known. There are 48 cards eligible to be the smallestcard in a straight flush. Hence, there are 48 straight flushes.

In forming a 3-of-a-kind hand, there are 13 choices for the rank and4 choices for the 3 cards of the given rank. This implies there are3-of-a-kind hands.

The ranks of the cards in a straight have the form x,x+1,x+2, wherex can be any of 12 ranks. There are then 4 choices for each card ofthe given ranks. This yields total choices. However,this count includes the straight flushes. Removing the 48 straightflushes leaves us with 720 straights.

To count the number of flushes, we obtain choicesfor 3 cards in the same suit. Of these, 12 are straight flushes whoseremoval leaves 274 flushes of a given suit. Multiplying by 4 produces1,096 flushes.

Total Number Of Poker Hands

Now we count the number of hands with a pair. There are 13 choices forthe rank of the pair, and pairs of the chosen rank.The non-pair card can be any of the remaining 48. Thus, thereare 3-card hands with a single pair.

We could determine the number of high card hands by removing the handswhich have already been counted in one of the previous categories.Instead, let us count them independently and see if the numbers sumto 22,100 which will serve as a check on our arithmetic.

A high card hand has 3 distinct ranks, but does not allow ranks of theform x,x+1,x+2 as that would constitute a straight. Thus, there arepossible sets of ranks from which we remove the12 sets of the form .This leaves 274 sets of ranks.For a given set of ranks, there are 4 choices for each cardexcept we cannot choose all in the same suit. Hence, there are274(4³-4) = 16,440 high card hands.

Number Of Two Pair Poker Hands

If we sum the preceding numbers, we obtain 22,100 and we can be confidentthe numbers are correct.

Here is a table summarizing the number of 3-card poker hands. Theprobability is the probability of having the hand dealt to you whendealt 3 cards.

Number Of Five Card Poker Hands