   Chapter 9.2, Problem 20ES ### 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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# Modify Example 9.2.4 by supposing that a PIN must not begin with any of the letters A—M and must end with a digit. Continue to assume that no symbol may be used more than once and that the total number of PINs is to be determined. a. Find the error in the following “solution.” “Constructing a PIN is a four-step process. Step 1: Choose the left-most symbol. Step 2: Choose the second symbol from the left. Step 3: Choose the third symbol from the left. Step 4: Choose the right-most symbol. Because none of the thirteen letters from A through M may be chosen in step 1, there are 36 − 13 = 23 ways to perform step 1. There are 35 ways to perform step 2 and 34 ways to perform step 3 because previously used symbols may not be used. Since the symbol chosen in step 4 must be a previously unused digit, there are 10 − 3 = 7 ways to perform step 4. Thus there are 23 ⋅ 35 ⋅ 34 ⋅ 7 = 191 , 590 different PINs that satisfy the given conditions.” b. Reorder steps 1—4 in part (a) as follows: Step 1: Choose the right-most symbol.Step 2: Choose the left-most symbol. Step 3: Choose the second symbol from the left. Step 4: Choose the third symbol from the left. Use the multiplication rule to find the number of PINs that satisfy the given conditions.

To determine

(a)

To find:

The error of the given solution to construct a PIN in a four-step process.

Explanation

Given information:

There are four digits in the PIN. The PIN does not startwith any letter between AM and the last symbol should be a number without any repeated symbols.

A process to select a PIN is given by following four steps.

Step 1: Choose the left-most symbol.

Step 2: Choose the second symbol from the left.

Step 3: Choose the third symbol from the left.

Step 4: Choose the right-most symbol.

By following the above four steps,

There are 3613=23 ways to choose the first symbol without any letter between AM. Without first chosen symbol, there are 35 ways to select the second symbol. Then the third symbol can be chosen in 34 ways. Because the last symbol must be a number, the fourth step can be performed in 103=7 ways. Therefore, by the multiplication rule, there are 23×35×34×7=191,590 different ways to choose a PIN

To determine

(b)

To find:

The number of PINs that satisfy the conditions by following the process given.

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