CSCE 235
Handout
21: Poker Cards
Assigned March 12, 2004
A deck of cards consists of four suits called clubs, diamonds, hearts, and spades. Each suit consists of thirteen cards with values A, 2, 3, 4, 5, 6,7, 8, 9, 10, J, Q, K. A poker hand is a set of 5 cards from a 52-card deck of cards. The order in which the cards are chosen is irrelevant. A straight consists of five cards whose values form a consecutive sequence such 8, 9, 10, J, Q. The ace A can be at the bottom of a sequence A, 2, 3, 4, 5, or at the top of a sequence, 10, J, Q, K, A. Poker hands are classified into disjoint sets as follows: they are listed in reverse order of their likelihood:
Royal Flush: 10, J, Q, K, A all in the same suit
Straight Flush: A straight all in the same suit that is not a royal flush
Four of a kind: Four cards in the hand have the same value. For example, four 3s and a 9.
Full house: Three cards of one value and two cards of another value.
Flush: Five cards all in the same suit, but not a royal or straight flush.
Straight: A straight that is not a royal or a straight flush.
Three of a kind: Three cards of one value, a fourth card of a second value, and a fifth card of a third value.
Two pairs: Two cards of one value, two more cards of a second value and the remaining card a third value.
One pair: Two cards of one value, but not classified above.
Nothing: None of the above.
(a) How many possible poker hands are there?
.
(b) How
many poker hands are full houses? Let’s
call a hand consisting of three jacks and two 8s a full house of type 
, with similar notation for other types of full houses. Order matters, since hands of type 
have three 8s and two
jacks. Also, types like 
and 
are impossible. So, types of full houses correspond to
2-permutations of the set of possible values of cards; hence, there are
(13)(12) different types of full houses.
Now
we count the number of full houses of each type, say type 
. There are 
ways to choose three
jacks from four jacks, and then there are 
ways to choose two 8s
from four 8s. Thus, there are (4)(6) =
24 hands of type 
. So, there are
(13)(12)(24) = 3744 full houses.
(c)
How many poker hands are two pairs? Let’s say that a hand with two pairs is of type 
if it consists of two
queens and two 4s. This time, we have
used set notation because order does not matter: hands of type 
are hands of type 
and we do not want to
count them twice. There are thus 
types of hands. For each type, say 
, there are 
of choosing two
queens, 
ways of choosing two
4s. And do not forget about the fifth
card. There are 52 – 8 = 44 ways of
choosing the fifth card. Hence there
are
poker hands consisting of two pairs.
(d)
How many poker hands are straights? First we count all possible straights even if they are royal or
straight flushes. Let’s call a straight
consisting of the values 8, 9, 10, J, Q a straight of type Q. In general, the type of a straight is the
highest value in the straight. Since,
any of the values 5, 6, 7, 8, 9, 10, J, Q, K, A can be the highest value in a
straight, there are 10 types of straights.
Given a type of straight, there are 4 choices for each of the 5
values. So, there are 
straights of each type and 
straights all
together.
There are 4 royal flushes and 36 straight flushes (why?) and so there are 10,200 straights that are not of these exotic varieties.
Please complete the following to find out why the answer is correct.
(e) How many poker hands are four of a kinds?
. (Why?)
(f) How many poker hands are flushes (not counting straight or royal flushes)?
. (Why?)
(g) How many poker hands are three of a kinds?
. (Why?)
(h) How many poker hands are one pairs?
1,098,240. (Why?)
Problems
based on (Ross and Wright 1988).