(a) How many passwords are possible? (b) How many passwords are possible that do NOT repeat any letters or numbers? (c) How many passwords are possible that have at least one character repeated?
(a) How many passwords are possible? (b) How many passwords are possible that do NOT repeat any letters or numbers? (c) How many passwords are possible that have at least one character repeated?
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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![**Understanding Password Combinations**
**Problem Statement:**
Each character in a password of length six can be any uppercase letter from A-Z or any number 0-9. For this problem, you may leave your answers unreduced.
**Questions:**
**(a) How many passwords are possible?**
**(b) How many passwords are possible that do NOT repeat any letters or numbers?**
**(c) How many passwords are possible that have at least one character repeated?**
**Explanation:**
Let's break down how to approach each part of the problem:
**Part (a):**
*How many passwords are possible?*
Since each character in the password can be any of the 26 uppercase letters or 10 digits, there are a total of \(26 + 10 = 36\) possible characters for each slot in the password. Each of the six characters can be chosen independently.
Thus, the total number of possible passwords is:
\[ 36^6 \]
**Part (b):**
*How many passwords are possible that do NOT repeat any letters or numbers?*
For a six-character password with no repeats, you have 36 choices for the first character, 35 choices for the second character, 34 choices for the third character, and so on.
Thus, the total number of possible passwords with no repeated characters is given by the permutation:
\[ 36 \times 35 \times 34 \times 33 \times 32 \times 31 \]
**Part (c):**
*How many passwords are possible that have at least one character repeated?*
One way to approach this is to subtract the number of passwords with no repeats (determined in part (b)) from the total number of passwords (determined in part (a)).
So, the number of passwords that have at least one repeated character is:
\[ 36^6 - (36 \times 35 \times 34 \times 33 \times 32 \times 31) \]
Understanding these principles helps in grasping combinatorial calculations often used in password security assessments.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe7646cd-9dad-44ff-a49d-6799ce85d2db%2F860360f3-46b0-4884-bc89-ee2a5415c1d8%2Ftdcym6k_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Password Combinations**
**Problem Statement:**
Each character in a password of length six can be any uppercase letter from A-Z or any number 0-9. For this problem, you may leave your answers unreduced.
**Questions:**
**(a) How many passwords are possible?**
**(b) How many passwords are possible that do NOT repeat any letters or numbers?**
**(c) How many passwords are possible that have at least one character repeated?**
**Explanation:**
Let's break down how to approach each part of the problem:
**Part (a):**
*How many passwords are possible?*
Since each character in the password can be any of the 26 uppercase letters or 10 digits, there are a total of \(26 + 10 = 36\) possible characters for each slot in the password. Each of the six characters can be chosen independently.
Thus, the total number of possible passwords is:
\[ 36^6 \]
**Part (b):**
*How many passwords are possible that do NOT repeat any letters or numbers?*
For a six-character password with no repeats, you have 36 choices for the first character, 35 choices for the second character, 34 choices for the third character, and so on.
Thus, the total number of possible passwords with no repeated characters is given by the permutation:
\[ 36 \times 35 \times 34 \times 33 \times 32 \times 31 \]
**Part (c):**
*How many passwords are possible that have at least one character repeated?*
One way to approach this is to subtract the number of passwords with no repeats (determined in part (b)) from the total number of passwords (determined in part (a)).
So, the number of passwords that have at least one repeated character is:
\[ 36^6 - (36 \times 35 \times 34 \times 33 \times 32 \times 31) \]
Understanding these principles helps in grasping combinatorial calculations often used in password security assessments.
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