Consider the problem "A common type of bike lock requires a six digit password, and only allows digits O through 9. How many passwords are there that contain the digit 9?" A student reasons: There are 6 possible places to put the 9, and then no restriction. So there are 105 possible arrangements of the other digits. This means the correct answer is 6 · 105. This solution is incorrect, since the student overcounts passwords containing more than one 1. This solution is incorrect, since, in the first step, the student needs to specify which digit gets the one. This solution is incorrect since there are not 10x10x10x10x10 arrangements of the digits O through 8 in the other three places. This solution is correct.

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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**Educational Example: Counting Passwords Containing a Specific Digit**

**Problem Statement:**
Consider the problem "A common type of bike lock requires a six-digit password, and only allows digits 0 through 9. How many passwords are there that contain the digit 9?"

**Student's Reasoning:**
A student reasons: There are 6 possible places to put the 9, and then no restriction. So there are \(10^5\) possible arrangements of the other digits. This means the correct answer is \(6 \cdot 10^5\).

**Multiple Choice Answers:**

- **A.** This solution is incorrect, since the student overcounts passwords containing more than one 1.
  
- **B.** This solution is incorrect, since, in the first step, the student needs to specify which digit gets the one.
  
- **C.** This solution is incorrect since there are not \(10 \times 10 \times 10 \times 10 \times 10 \times 10\) arrangements of the digits 0 through 8 in the other three places.
  
- **D.** This solution is correct.

**Explanation of the Options:**

- **Option A** suggests a misunderstanding about overcounting certain passwords. It assumes a misstep in the counting process, but it doesn't identify a correct counting principle.

- **Option B** points out that the student's reasoning might need clarification in identifying which digit gets the 9, which is not relevant in this context.

- **Option C** indicates a misunderstanding by claiming there are no arrangements of \(10:10:10:10:10\) for the remaining digits. However, this concept does not correctly analyze the problem.

- **Option D** correctly identifies that the student's logic was sound. There are 6 possible positions for the digit 9, and for the remaining positions, there are \(10^5\) possible arrangements of the other digits.

This problem illustrates the importance of understanding combinatorial principles in counting distinct possibilities in password formation.
Transcribed Image Text:**Educational Example: Counting Passwords Containing a Specific Digit** **Problem Statement:** Consider the problem "A common type of bike lock requires a six-digit password, and only allows digits 0 through 9. How many passwords are there that contain the digit 9?" **Student's Reasoning:** A student reasons: There are 6 possible places to put the 9, and then no restriction. So there are \(10^5\) possible arrangements of the other digits. This means the correct answer is \(6 \cdot 10^5\). **Multiple Choice Answers:** - **A.** This solution is incorrect, since the student overcounts passwords containing more than one 1. - **B.** This solution is incorrect, since, in the first step, the student needs to specify which digit gets the one. - **C.** This solution is incorrect since there are not \(10 \times 10 \times 10 \times 10 \times 10 \times 10\) arrangements of the digits 0 through 8 in the other three places. - **D.** This solution is correct. **Explanation of the Options:** - **Option A** suggests a misunderstanding about overcounting certain passwords. It assumes a misstep in the counting process, but it doesn't identify a correct counting principle. - **Option B** points out that the student's reasoning might need clarification in identifying which digit gets the 9, which is not relevant in this context. - **Option C** indicates a misunderstanding by claiming there are no arrangements of \(10:10:10:10:10\) for the remaining digits. However, this concept does not correctly analyze the problem. - **Option D** correctly identifies that the student's logic was sound. There are 6 possible positions for the digit 9, and for the remaining positions, there are \(10^5\) possible arrangements of the other digits. This problem illustrates the importance of understanding combinatorial principles in counting distinct possibilities in password formation.
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