# Number Of Possible Poker Hands

**Brian Alspach**

**13 January 2000**

In this video we will go over the number of ways of getting the most common poker or sought poker hands using combinations and the multiplication principle. SOLUTION: Again, the number of possible poker hands is 52 C 5 = 52 51 50 49 48 / 5! How many poker hands contain exactly four aces? Such a hand consists of 4 aces (chosen from the 4 aces in the deck) together with 1 non-ace (chosen from the 48 other cards in the deck).

### Abstract:

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. One factor is whether 'the number of hands' is the number of possible hands a single person can have, or whether it's the number of possible overall board states (that is, taking into account every player's hand). If the latter, we would have to know the number of players. – Acccumulation Aug 26 at 4:45. There are Combination (52,5) = 2.598.960 poker hands in a deck.Know that this number includes all possible hands. Hand Combination LikelyStraight flush 40 0.00154%Four of a kind 624 0.02401%Full.

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 .

## Number Of Possible Poker Hands Drawn

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.

## Number Of Possible Hands In Poker

The ranks of the cards in a straight have the form *x*,*x*+1,*x*+2, where*x* 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.

## Find The Total Number Of Possible 5 Card Poker Hands

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.

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(4^{3}-4) = 16,440 high card 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.

## List Of Best Poker Hands

hand | number | Probability |

straight flush | 48 | .0022 |

3-of-a-kind | 52 | .0024 |

straight | 720 | .0326 |

flush | 1,096 | .0496 |

pair | 3,744 | .1694 |

high card | 16,440 | .7439 |

## Number Of Possible Poker Hands Spread

last updated 13 January 2000