   Chapter 9.5, Problem 30ES ### 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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# Find the error in the following reasoning: “Consider forming a poker hand with two pairs as a five-step process.”Step 1: Choose the denomination of one of the pairs.Step 2: Choose the two cards of that denomination.Step 3: Choose the denomination of the other of the pairs.Step 4: Choose the two cards of that second denomination.Step 5: Choose the fifth card from the remaining denominations.There are ( 13 1 ) ways to perform step 1, ( 4 2 ) ways to perform step 2, ( 12 1 ) ways to perform step 3, ( 4 2 ) ways to perform step 4, and ( 44 1 ) ways to perform step 5. Therefore, the total number of five-card poker hands with two pairs is 13 ⋅ 6 ⋅ 12 ⋅ 6 ⋅ 44 = 247.104. "

To determine

To find out the error in the given reasoning.

Explanation

Given information:

“Consider forming a poker hand with two pairs as a five-step process.

Step 1- Choose the denomination of one of the pairs.

Step 2- Choose the two cards of that denomination.

Step 3- choose the denomination of the other of pairs.

Step 4- choose the two cards of that second denomination.

Step 5- choose the fifth card from the remaining denominations.

There are (131) ways to perform step 1, (42) ways to perform the step 2, (121) ways to perform step 3, (42) ways to perform step 4, and (441) ways to perform step 5. Therefore the total number of five-card poker hands with two pairs is 13612644=247,104 .”

Calculation:

To form a poker hand with two pairs as a five-step process can be done as follows-

There are total 13 denominations in a deck of 52 cards. There are total 4 cards in each denomination.

The question is about forming a poker hand with two pairs. So in the Step 1 only, the two pairs of the denomination should be chosen.

Step 1- Choosing the denomination of one of the pairs. Since there are 13 denominations, so choosing 1 can be done in (131) ways but there is a need to choose the two pairs not one pair so this step is incorrect. It should have been choosing two pairs from 13 denominations in (132) ways.

Step2- Since there are total 4 cards in each denomination. So choosing the two cards from the same denomination can be done in (42) ways.So the step 2 is correct

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