Pascal's Triangle - How many poker hands?

		Main

	Applications		Identities

History		FAQ		Algorithms

How many different poker hands are there?

This is a question that has several different answers; let’s start with one valid mathematical answer and show how it compares to the usual understanding of the rules of poker.

If we take a standard deck of 52 cards (no joker) and consider a poker hand to be a collection of 5 cards out of that 52, this would say there are

= 2,598,960 different combinations of five cards where the order in which they were dealt is unimportant. To look at the rules of poker, the most generalized description of poker hands would say there are only 9 different kinds of hands: straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, one pair and no pair. Obviously, these numbers are far apart and need to be reconciled.

Poker players will realize that “straight” or “straight flush” actually describes many hands that are not equal in a showdown, to take one example. The usual way to declare a straight or straight flush is to also mention the highest card in the sequence by saying “jack high straight” (or “straight flush to the jack in spades”) to signify the hand JS-10H-9S-8H-7D (or JS-10S-9S-8S-7S); in most variations of poker, the ace is the only card that counts both as the highest card and the lowest in two different sequences; “ace high straight” is the card sequence A-K-Q-J-10 (known as a royal flush if all the cards are of the same suit) and a “five high straight” is 5-4-3-2-A (known as a “bicycle” or a “wheel”, the best possible hand in a poker variant called Lowball or in high-low split games.) This means there are actually 10 different straights and 10 different straight flushes in terms of value as poker hands, since of all thirteen different denominations, only the 4, 3 and 2 can’t be the highest card in a straight. In terms of math, even “jack high straight flush” still describes four different hands, since any suit can be involved, and “jack high straight” describes 4⁵ – 4 = 1,020 hands, which is all the possible combinations of suits for all five card removing the four even “jack high straight flushes” we already counted.

For the first step, let’s look at poker hands in terms of how many common cards they have: four of a kind, full house, three of a kind, two pair, one pair and no pair; no pair will include straights, flushes and straight flushes for the time being. We will count how many different five card combinations are collected together under a single poker hand name, such as “four of a kind” or “two pair”.

Type of poker hand	# of five card combinations in type
four of a kind	= 624
full house	= 3744
three of a kind	= 54912
two pair	= 123552
one pair	= 1098240
no pair *	= 1317888
TOTAL	2598960

Notice that, so far, if a hand is higher on the list, it is rarer; if someone says, “I have a pair”, there are 1,098,240 different combinations of five cards that satisfy this phrase. There are

= 13 different choices for the denomination of the pair, and when you get a pair of jacks, for instance, there are

= 6 different ways to do that.: JH-JD, JH-JC, JH-JS, JD-JC, JD-JS, JC-JS. If you have a pair you also have three more cards that do not match the pair or each other, known as “kickers” in poker terms; these can be useful for breaking ties if two players both have a pair of jacks. There are

= 220 different three card combinations once some pair has been chosen, and each specific three card combo, there are 43 different combinations possible, since if I say “pair of jacks with king, 7, 3 as kickers”, there are four possible kings, four possible 7s and four possible 3s.

Here are the number of each of the nine major types of poker hands there are, breaking out the valuable “no pair” hands (straight flushes, straights and flushes) from the bad no pair hands. Notice that in both the flush and straight totals, we subtract 40; this is because technically, a straight flush is both a straight and a flush, but we’ve already counted the straight flushes in their own category, and we don’t want to count them twice; likewise, the – 15,348 in the no pair category is due to the removal of straights, flushes and straight flushes.

Type of poker hand	# of five card combinations in type
Straight flush	10×4 = 40
four of a kind	= 624
full house	= 3744
Flush	- 40 = 5108
Straight	10×4⁵ – 40 = 10200
three of a kind	= 54912
two pair	= 123552
one pair	= 1098240
no pair - straight and flushes	- 15348 = 1302540
TOTAL	2598960

Questions for the reader.

1. With both four of a kind and full house,

is in the formula for number of hands, but with two pair, we use

; why the difference?

2. There are two players left in a poker hand, Megan and Luis; Megan declares “Eight high straight” and shows 8D-7C-6C-5D-4H; Luis says “Me, too” before he shows his cards. Assuming Luis is telling the truth and Megan hasn’t seen any cards other than the five in her hand, how many different combinations of cards can Luis be holding? (Warning: watch out for straight flushes.)

3. In a popular variant of poker called Texas Hold ‘Em, each player has two cards in his or her hand, while five community cards (known as the board) are dealt face up in the middle; the winner is the player who can make the best five card hand out of the seven he or she can see; any player staying in until the showdown is allowed to use any combination of community and hand cards, even if other players are using the same community cards to make their best hands. The five community cards are AD-JC-7S-5H-2C and your hand is AH-JH. How many two card combinations can beat you? How many can tie?

4. As we said above, the 40 straight flushes could be counted as 10 different valued classes of hands, with 4 hands appearing in each class.
a. Split up each of the nine major types of hands into different valued classes and count the number of hands that appear in each class.

b. Now that we know the different value classes, which poker hands, would constitute the top 10% classes of hands? Which poker hands would constitute the top 1% of hands?