   Chapter 9.5, Problem 11ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Refer to Example 9.5.9. For each poker holding below, (1) find the number of five-card poker hands with that holding; (2) find the probability that a randomly chosen set of five cards has that holding. royal flush straight flush four of a kind full house flush straight (including a straight flush and a royal flush) three of a kind one pair neither a repeated denomination not five of the same suit nor five adjacent denominations

To determine

(a)

To find the number of five-card poker handswith the given holding.

Explanation

Given information:

The given holdings are

1. Royal flush
2. Straight flush
3. Four of a kind
4. Full house
5. Flush
6. Straight (including a straight flush and a royal flush).
7. Three of a kind
8. One pair
9. Neither a separated denomination nor five of the same suit nor five adjacent denominations.

Calculation:

There are five cards holding in the game of poker with some special names. These are as follows-

a)Royal flush: 10,J,Q,K,A of the same suit.

It is known that in a deck of 52 cards, there are 4 suits and each suit consists of a royal flush.

Therefore the number of royal flush poker hands available =4

b) Straight flush: the following are the straight flushes from high to low A1234,1245,23456,34567,45678,56789,678910,78910J,8910JQ,910JQK,10JKQA except 10,J,Q,K,A i.e. royal flush.

So there are total 9 straight flushed of each suit.

It is known that in a deck of 52 cards, there are 4 suits and each suit consists of 9 straight flushes.

Therefore the total number of straight flushes in five card poker hands =9×4=36

c) Four of a kind: selecting four of a kind means selecting four cards of the same denomination. So there are total 13 possibilities and 5 th card from the remaining 48 in 48 ways.

Therefore the number of four of a kind available in five card-poker hands available =13×48=624

d)Full house: Full house comprises of 3 cards of one denomination and it is known that in a deck of 52 cards, there are total 4 suits and each suit consists of a 13 denominations only one denomination need to be chosen.

Therefore the number of five-card poker hands with full house is = choosing 3 cards × choosing remaining two cards of same denomination.

=( 4 3)( 13 1)×( 4 2)×( 12 1)=4×13×6×12=52×72=3744

Therefore the number of poker hands with full house is =3744

e)Flush: a flush is a set of cards from the same suit that doesn’t contain the straight flush and royal flush

To determine

(b)

To find the number of five-card poker hands with the given holding.

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