Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Password Combinations and Probabilities
#### Problem Statement
A website requires users to create a 4-character long password. Each character can be a number, or a symbol chosen from ?, #, !, and $. The password does not have to include both numbers and symbols.
##### Questions:
a) How many 4-character passwords are possible if numbers and symbols can be used more than once?
b) How many 4-character passwords are possible if no numbers or symbols can be repeated?
c) What is the probability that a randomly generated 4-character password contains 4 different characters?
---
#### Solution:
---
**Part (a)**: Set up the expression that can be used to calculate the number of possible 4-character passwords if numbers and symbols can be used more than once.
- The expression is \[14^4\]. (Do not simplify.)
Here, 14 represents the total number of possible characters (10 digits + 4 symbols).
- The number of different passwords that are possible if numbers and symbols can be used more than once is \[38416\].
---
**Part (b)**: Determine the number of different passwords that are possible if no numbers or symbols can be repeated.
- Using the principle of permutation:
\[ 14 \times 13 \times 12 \times 11 \]
- Simplifying the expression:
\[ 24024 \]
---
**Part (c)**: Calculate the probability that a 4-character password contains 4 different characters.
- The probability is the number of favorable outcomes divided by the total number of outcomes.
- Probability = \[ \frac{24024}{38416} \approx 0.626 \]
In summary, the probability that a randomly generated 4-character password contains 4 different characters is approximately 0.626 (rounded to three decimal places).](https://content.bartleby.com/qna-images/question/7b1d7e88-5924-4190-9ef7-53bc83ebfef1/186240b5-d2dc-4cf5-a0e3-2d6b94966fdf/rgy8gnq_processed.png)
Transcribed Image Text:### Password Combinations and Probabilities
#### Problem Statement
A website requires users to create a 4-character long password. Each character can be a number, or a symbol chosen from ?, #, !, and $. The password does not have to include both numbers and symbols.
##### Questions:
a) How many 4-character passwords are possible if numbers and symbols can be used more than once?
b) How many 4-character passwords are possible if no numbers or symbols can be repeated?
c) What is the probability that a randomly generated 4-character password contains 4 different characters?
---
#### Solution:
---
**Part (a)**: Set up the expression that can be used to calculate the number of possible 4-character passwords if numbers and symbols can be used more than once.
- The expression is \[14^4\]. (Do not simplify.)
Here, 14 represents the total number of possible characters (10 digits + 4 symbols).
- The number of different passwords that are possible if numbers and symbols can be used more than once is \[38416\].
---
**Part (b)**: Determine the number of different passwords that are possible if no numbers or symbols can be repeated.
- Using the principle of permutation:
\[ 14 \times 13 \times 12 \times 11 \]
- Simplifying the expression:
\[ 24024 \]
---
**Part (c)**: Calculate the probability that a 4-character password contains 4 different characters.
- The probability is the number of favorable outcomes divided by the total number of outcomes.
- Probability = \[ \frac{24024}{38416} \approx 0.626 \]
In summary, the probability that a randomly generated 4-character password contains 4 different characters is approximately 0.626 (rounded to three decimal places).
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