A computer password consists of six characters.
a. How many different passwords are possible if each character may be any lowercase letter or digit?
b. How many different passwords are possible if each character may be any lowercase letter or digit, and at least one character must be a digit?
c. A computer system requires that passwords contain at least one digit. If seven characters are generated at random, and each is equally likely to be any of the 26 letters or 10 digits, what is the probability that a valid password will be generated?
The Correct Answer and Explanation is :
a. Number of different passwords possible (letters or digits)
For a password consisting of 6 characters, each character can be either a lowercase letter (26 choices) or a digit (10 choices). Therefore, each character has ( 26 + 10 = 36 ) possible choices.
Since there are 6 characters, the total number of different passwords is:
[
36^6 = 2,176,782,336
]
b. Number of different passwords possible (at least one digit)
In this case, we need to count the total number of passwords and subtract those that consist only of letters.
- Total number of passwords (letters or digits): As calculated in part (a), the total number of possible passwords is ( 36^6 = 2,176,782,336 ).
- Number of passwords with only letters: If all six characters are letters, then there are only 26 possible choices for each character. Therefore, the number of passwords with only letters is:
[
26^6 = 308,915,776
]
- Passwords with at least one digit: The number of passwords with at least one digit is the total number of passwords minus the number of passwords with only letters:
[
36^6 – 26^6 = 2,176,782,336 – 308,915,776 = 1,867,866,560
]
c. Probability of generating a valid password (at least one digit)
Here, we are asked to find the probability of generating a valid password (at least one digit) with 7 characters, where each character is equally likely to be a letter or a digit.
- Total possible passwords: For 7 characters, each can be a letter or a digit, so there are ( 36 ) choices per character. Thus, the total number of possible passwords is:
[
36^7 = 78,364,164,096
]
- Passwords with only letters: If all 7 characters are letters, then there are 26 choices for each character. Therefore, the number of passwords with only letters is:
[
26^7 = 8,031,810,176
]
- Passwords with at least one digit: The number of valid passwords (with at least one digit) is the total number of passwords minus the number of passwords with only letters:
[
36^7 – 26^7 = 78,364,164,096 – 8,031,810,176 = 70,332,353,920
]
- Probability of a valid password: The probability that a password contains at least one digit is the ratio of valid passwords to total passwords:
[
\frac{70,332,353,920}{78,364,164,096} \approx 0.897
]
Thus, the probability that a randomly generated 7-character password contains at least one digit is approximately 0.897, or 89.7%.
Summary:
- a. ( 36^6 = 2,176,782,336 ) passwords are possible.
- b. ( 1,867,866,560 ) passwords have at least one digit.
- c. The probability that a valid password (at least one digit) is generated is approximately 0.897 or 89.7%.
These calculations highlight the enormous number of possible passwords and the importance of including at least one digit for security and system requirements.