4-Card Poker Hands

We determine the number of 4-card poker hands.

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 ${{52}\choose{4}}= 270,725$ .

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 $4\cdot 13\cdot 48 = 2,496$ 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 $4^4\cdot 11=2,816$ 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 ${{13}\choose{2}} = 78$ choices for the two ranks of the pairs. There are ${{4}\choose{2}}=6$ choices for each of the pairs. This produces $6\cdot 6\cdot 78 = 2,808$ hands of two pairs.

To count the number of flushes, we obtain ${{13}\choose{4}}=715$ 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 ${{12}\choose{2}} = 66$ choices for the ranks of the other 2 cards and 4 choices for each of these 2 cards. We have $13\cdot 6\cdot 66\cdot 16 = 82,368$ 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 ${{13}\choose{4}}=715$ possible sets of ranks from which we remove the 11 sets of the form $\{x,x+1,x+2,x+3\}$ . This leaves 704 sets of ranks. For a given set $\{w,x,y,z\}$ of ranks, there are 4 choices for each card except we cannot choose all in the same suit. Hence, there are 704(4⁴-4) = 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

4-Card Poker Hands

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