**Problem Statement:**Pamela DeMar's computer printer allows for optional settings with a panel of five on-off switches in a row. How many different settings can she select if no two adjacent switches can both be on?**Solution:**To solve this problem, we'll consider the restrictions and calculate the number of possible settings.We need to determine the number of ways to arrange the switches such that no two adjacent switches are both on.Define:- $ S_0 $: Number of arrangements with no switches on.- $ S_1 $: Number of arrangements with exactly one switch on.- $ S_2 $: Number of arrangements with exactly two switches on.- $ S_3 $: Number of arrangements with exactly three switches on.- $ S_4 $: Number of arrangements with exactly four switches on.- $ S_5 $: Number of arrangements with exactly five switches on. Note, $ S_5 = 0 $ because no two on-switches can be adjacent.Let's use a recursive approach to find the count of valid arrangements for $ n $ switches:For $ n = 0 $, clearly, there is only one way (do nothing). So, $ S_0 = 1 $.For $ n = 1 $, there are two configurations: off, on. So, $ S_1 = 2 $.For $ n = 2 $, the configurations are: off-off, off-on, on-off. So, $ S_2 = 3 $.For $ n = 3 $, the configurations include patterns of $ n=2 $ followed by off: off-off-off, off-off-on, off-on-off, on-off-off, on-off-on. So, by this recursive pattern, continue increasing.**Calculate for $ n = 5 $:**1. Arrange switch configurations for 3 switches and follow by: off -> 4 valid configurations.2. Arrange switch configurations for 2 switches and follow by: on-off -> 3 valid configurations.3. Arrange switch configurations for 1 switch and follow by: on-off-off -> 2 valid configurations.Hence, $ S_5 ≈ 8 $.**Conclusion:**The total number of possible settings where no two adjacent switches are on is $ S_5 = 8 $. **Answer:** Enter the number **8** in the answer box.

There are 5 switches with each switch either in on or off state. Total no of settings = 25 = 32 Let 0 represent off state and 1 represent on state The different settings are: 00000, 00001, 00010, 00011, 00100, 00101, 00110, 00111, 01000, 01001, 01010, 01011, 01100, 01101, 01110, 01111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111 Settings in which no two adjacent switch is both on = 00000, 00001, 01010, 00100, 00101, 01000, 01001, 01010, 10000, 10001, 10010, 10100, 10101 No of settings in which no two adjacent settings is both On = 13

Math

Probability

Pamela DeMar's computer printer allows for optional settings with a panel of five on-off switches in a row. How many different settings can she select if no two adjacent switches can both be on? The number of possible settings is (Type a whole number.)

Pamela DeMar's computer printer allows for optional settings with a panel of five on-off switches in a row. How many different settings can she select if no two adjacent switches can both be on? The number of possible settings is (Type a whole number.)

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

**Problem Statement:**

Pamela DeMar's computer printer allows for optional settings with a panel of five on-off switches in a row. How many different settings can she select if no two adjacent switches can both be on?

**Solution:**

To solve this problem, we'll consider the restrictions and calculate the number of possible settings.

We need to determine the number of ways to arrange the switches such that no two adjacent switches are both on.

Define:
- $ S_0 $: Number of arrangements with no switches on.
- $ S_1 $: Number of arrangements with exactly one switch on.
- $ S_2 $: Number of arrangements with exactly two switches on.
- $ S_3 $: Number of arrangements with exactly three switches on.
- $ S_4 $: Number of arrangements with exactly four switches on.
- $ S_5 $: Number of arrangements with exactly five switches on. Note, $ S_5 = 0 $ because no two on-switches can be adjacent.

Let's use a recursive approach to find the count of valid arrangements for $ n $ switches:

For $ n = 0 $, clearly, there is only one way (do nothing). So, $ S_0 = 1 $.

For $ n = 1 $, there are two configurations: off, on. So, $ S_1 = 2 $.

For $ n = 2 $, the configurations are: off-off, off-on, on-off. So, $ S_2 = 3 $.

For $ n = 3 $, the configurations include patterns of $ n=2 $ followed by off: off-off-off, off-off-on, off-on-off, on-off-off, on-off-on. So, by this recursive pattern, continue increasing.

**Calculate for $ n = 5 $:**

1. Arrange switch configurations for 3 switches and follow by: off -> 4 valid configurations.

2. Arrange switch configurations for 2 switches and follow by: on-off -> 3 valid configurations.

3. Arrange switch configurations for 1 switch and follow by: on-off-off -> 2 valid configurations.

Hence, $ S_5 ≈ 8 $.

**Conclusion:**

The total number of possible settings where no two adjacent switches are on is $ S_5 = 8 $.

**Answer:** Enter the number **8** in the answer box.

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