Counting

The types of 5-card poker hands are

Another important mathematical tool we can use is the number of total hands possible of a certain rank. Here we show how many of each rank of poker hand there are and how we compute these numbers. The total number of 5-card poker hands is ${{52}\choose{5}}= 2,598,960$.

A straight flush is completely determined once the smallest card in the straight flush is known. There are 36 cards eligible to be the smallest card in a straight flush. Every card but the 10,J,Q,K of every suit so 52-4*4=36. Hence, there are 36 straight flushes. Note we counted the 10 out since a 10 low straight flush is one of the 4 Royal flushes (A-K-Q-J-10 suited).

In forming a 4-of-a-kind hand, there are 13 choices for the rank of the quads, and 48 choices for the remaining card. This implies there are $48\cdot 13=624$ 4-of-a-kind hands.

There are 13 choices for the rank of the triple and 12 choices for the rank of the pair in a full house. There are C(4,3)=4 ways of choosing the triple of a given rank and C(4,2)=6 ways to choose the pair of the other rank. This produces $13\cdot 12\cdot 6\cdot 4=3,744$ full houses.

To count the number of flushes, we obtain ${{13}\choose{5}}=1,287$ choices for 5 cards in the same suit. Of these, 10 are straight flushes whose removal leaves 1,277 flushes of a given suit. Multiplying by the 4 suits produces 5,108 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 of the given ranks. This yields $4^5\cdot 10=10,240$ total choices. However, this count includes the straight flushes. Removing the 40 straight flushes leaves us with 10,200 straights.

In forming a 3-of-a-kind hand, there are 13 choices for the rank of the triple, and there are ${{12}\choose{2}}=66$ choices for the ranks of the other 2 cards. There are 4 choices for the triple of the given rank and there are 4 choices for each of the cards of the remaining 2 ranks. Altogether, we have $13\cdot 66\cdot 4^3=54,912$ 3-of-a-kind hands.

Next we consider two pairs hands. There are ${{13}\choose{2}} = 78$choices for the two ranks of the pairs. There are 6 choices for each of the pairs, and there are 44 choices for the remaining card. This produces $6^2\cdot 44\cdot 78 = 123,552$ hands of two pairs.

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 ${{12}\choose{3}} = 220$ choices for the ranks of the other 3 cards and 4 choices for each of these 3 cards. We have $13\cdot 6\cdot 220\cdot 4^3 = 1,098,240$ 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 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 the form x,x+1,x+2,x+3,x+4 as that would constitute a straight. Thus, there are ${{13}\choose{5}}=1,287$ possible sets of ranks from which we remove the 10 sets of the form $\{x,x+1,x+2,x+3,x+4\}$. This leaves 1,277 sets of ranks. For a given set $\{v,w,x,y,z\}$ of ranks, there are 4 choices for each card except we cannot choose all in the same suit. Hence, there are 1277(45-4) = 1,302,540 high card hands.

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

Here is a table summarizing the number of 5-card poker hands.

Hand Number
Royal Flush 4
Straight Flush 36
4-of-a-kind 624
Full House 3,744
Flush 5,108
Straight 10,200
3-of-a-kind 54,912
Two Pair 123,552
Pair 1,098,240
High Card 1,302,540
Total 2,598,960