## Poker Combinations Math

**Brian Alspach**

The probability of forming any given hand is the number of ways it can be arranged divided by the total number of combinations of 2,598.960. Below are the number of combinations for each hand. Just divide by 2,598,960 to get the probability. The next section shows how to derive the number of combinations of each poker hand in five. Hand combinations in poker all stem from statistics. So if you’re interested in finding out more about the math side of things, here are a few links that I found helpful: Combinations video - Youtube (all the stuff on this channel is awesome) Permutations and Combinations - MathForum.org. Poker Maths - Combinations What is Poker Combinatorics? Combinatorics is the practice of breaking down ranges and counting individual combinations of hands.

**13 January 2000**

- The formula in Excel is =COMBIN(52,3) Or in an online combinations calculator, you enter 52 as “n” and 3 as “r.” The answer is 22,100.
- The discussion is mostly mathematical, using the Poker hands to illustrate counting techniques and calculation of probabilities Working with poker hands is an excellent way to illustrate the counting techniques covered previously in this blog – multiplication principle, permutation and combination (also covered here).

### Abstract:

The types of 5-card poker hands are

- straight flush
- 4-of-a-kind
- full house
- flush
- straight
- 3-of-a-kind
- two pairs
- a pair
- high card

Most poker games are based on 5-card poker hands so the ranking ofthese hands is crucial. There can be some interesting situationsarising when the game involves choosing 5 cards from 6 or more cards,but in this case we are counting 5-card hands based on holding only5 cards. The total number of 5-card poker hands is.

A straight flush is completely determined once the smallest card in thestraight flush is known. There are 40 cards eligible to be the smallestcard in a straight flush. Extra slots ii rarity 12. Hence, there are 40 straight flushes.

In forming a 4-of-a-kind hand, there are 13 choices for the rank ofthe quads, 1 choice for the 4 cards of the given rank, and 48 choicesfor the remaining card. This implies there are 4-of-a-kind hands.

There are 13 choices for the rank of the triple and 12 choices for therank of the pair in a full house. There are 4 ways of choosing thetriple of a given rank and 6 ways to choose the pair of the other rank.This produces full houses.

To count the number of flushes, we obtain choicesfor 5 cards in the same suit. Of these, 10 are straight flushes whoseremoval leaves 1,277 flushes of a given suit. Multiplying by 4 produces5,108 flushes.

The ranks of the cards in a straight have the form *x*,*x*+1,*x*+2,*x*+3,*x*+4,where *x* can be any of 10 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 40 straightflushes leaves us with 10,200 straights.

In forming a 3-of-a-kind hand, there are 13 choices for the rank of thetriple, and there are choices for the ranks of theother 2 cards. There are 4 choices for the triple of the given rank andthere are 4 choices for each of the cards of the remaining 2 ranks.Altogether, we have 3-of-a-kind hands.

Next we consider two pairs hands. There are choices for the two ranks of the pairs. There are 6 choices for eachof the pairs, and there are 44 choices for the remaining card. Thisproduces hands of two pairs.

Now we count the number of hands with a pair. There are 13 choices forthe rank of the pair, and 6 choices for a pair of the chosen rank. Thereare choices for the ranks of the other 3 cardsand 4 choices for each of these 3 cards. We have hands with a 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 2,598,960 which will serve as a check on our arithmetic.

A high card hand has 5 distinct ranks, but does not allow ranks of theform *x*,*x*+1,*x*+2,*x*+3,*x*+4 as that would constitute a straight. Thus, thereare possible sets of ranks from which we remove the10 sets of the form .This leaves 1,277 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 are1277(4^{5}-4) = 1,302,540 high card hands.

If we sum the preceding numbers, we obtain 2,598,960 and we can be confidentthe numbers are correct.

Here is a table summarizing the number of 5-card poker hands. Theprobability is the probability of having the hand dealt to you whendealt 5 cards.

hand | number | Probability |

straight flush | 40 | .000015 |

4-of-a-kind | 624 | .00024 |

full house | 3,744 | .00144 |

flush | 5,108 | .0020 |

straight | 10,200 | .0039 |

3-of-a-kind | 54,912 | .0211 |

two pairs | 123,552 | .0475 |

pair | 1,098,240 | .4226 |

high card | 1,302,540 | .5012 |

### Poker Hand Combinations Math

last updated 12 January 2000