**Brian Alspach**

**13 January 2000**

We determine the number of 4-card poker hands.

The types of 4-card poker hands are

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

I actually do not know of any 4-card poker games, but given all the variations of poker which have been developed, it must be the case that people have played some 4-card games. Mambo stud does not count as it is a 4-card game but hands are based on 3-card poker hands contained in the 4 cards. The total number of 4-card poker hands is .

In forming a 4-of-a-kind hand, there are 13 choices for the rank and only 1 choice for the 4 cards of the given rank. This implies there are precisely 13 4-of-a-kind hands.

A straight flush is completely determined once the smallest card in the straight flush is known. There are 44 cards eligible to be the smallest card in a straight flush. Hence, there are 44 straight flushes.

In forming a 3-of-a-kind hand, there are 13 choices for the rank of the triple, there are 4 choices for 3 cards of a given rank, and there are 48 choices for the remaining card. Altogether, there are 3-of-a-kind hands.

The ranks of the cards in a straight have the form
*x*,*x*+1,*x*+2,*x*+3, where
*x* can be any of 11 ranks. There are then 4 choices for each card of
the given ranks. This yields
total choices. However,
this count includes the straight flushes. Removing the 44 straight
flushes leaves us with 2,772 straights.

Next we consider two pairs hands. There are choices for the two ranks of the pairs. There are choices for each of the pairs. This produces hands of two pairs.

To count the number of flushes, we obtain choices for 4 cards in the same suit. Of these, 11 are straight flushes whose removal leaves 704 flushes of a given suit. Multiplying by 4 produces 2,816 flushes.

Now we count the number of hands with a pair. There are 13 choices for the rank of the pair, and 6 choices for a pair of the chosen rank. There are choices for the ranks of the other 2 cards and 4 choices for each of these 2 cards. We have hands with a pair.

We could determine the number of high card hands by removing the hands which have already been counted in one of the previous categories. Instead, let us count them independently and see if the numbers sum to 270,725 which will serve as a check on our arithmetic.

A high card hand has 4 distinct ranks, but does not allow ranks of the
form
*x*,*x*+1,*x*+2,*x*+3 as that would constitute a straight. Thus, there are
possible sets of ranks from which we remove the
11 sets of the form
.
This leaves 704 sets of ranks.
For a given set
of ranks, there are 4 choices for each card
except we cannot choose all in the same suit. Hence, there are
704(4^{4}-4) = 177,408 high card hands.

If we sum the preceding numbers, we obtain 270,725 and we can be confident the numbers are correct.

Here is a table summarizing the number of 4-card poker hands. The probability is the probability of having the hand dealt to you when dealt 4 cards.

hand | number | Probability |

4-of-a-kind | 13 | .00005 |

straight flush | 44 | .00016 |

3-of-a-kind | 2,496 | .0092 |

straight | 2,772 | .0102 |

two pairs | 2,808 | .0104 |

flush | 2,816 | .0104 |

pair | 82,368 | .3042 |

high card | 177,408 | .6553 |

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