A computer password is 6-8 characters long. Each character must be a digit, an uppercase letter, or a lowercase letter. Each password must contain at least one digit, one uppercase letter and one lowercase letter. How many passwords are possible? Explain
A computer password is 6-8 characters long. Each character must be a digit, an
uppercase letter, or a lowercase letter. Each password must contain at least one digit,
one uppercase letter and one lowercase letter. How many passwords are possible?
Explain.
The Correct Answer and Explanation is :
To solve this problem, we need to find the number of possible passwords of length 6, 7, and 8 that contain at least one digit, one uppercase letter, and one lowercase letter. The characters available for the password can be a digit (0-9), an uppercase letter (A-Z), or a lowercase letter (a-z), so there are 10 + 26 + 26 = 62 possible characters to choose from for each position in the password.
Step 1: Total number of possible passwords (without restrictions)
For a password of length 6, 7, or 8, each character can be any of the 62 possible characters. Thus, the total number of possible passwords for each length is:
- For length 6: ( 62^6 )
- For length 7: ( 62^7 )
- For length 8: ( 62^8 )
Step 2: Subtract passwords that do not contain at least one digit, one uppercase letter, and one lowercase letter
We need to exclude the passwords that do not meet the requirement of containing at least one digit, one uppercase letter, and one lowercase letter. This involves using the principle of inclusion-exclusion.
Exclusion 1: Passwords without digits
If the password contains no digits, the 6, 7, or 8 characters must be selected from the 26 uppercase and 26 lowercase letters, so there are 52 choices per character. The number of passwords with no digits for each length is:
- For length 6: ( 52^6 )
- For length 7: ( 52^7 )
- For length 8: ( 52^8 )
Exclusion 2: Passwords without uppercase letters
If the password contains no uppercase letters, the characters must be selected from the 10 digits and 26 lowercase letters, giving 36 choices per character. The number of passwords with no uppercase letters for each length is:
- For length 6: ( 36^6 )
- For length 7: ( 36^7 )
- For length 8: ( 36^8 )
Exclusion 3: Passwords without lowercase letters
If the password contains no lowercase letters, the characters must be selected from the 10 digits and 26 uppercase letters, giving 36 choices per character. The number of passwords with no lowercase letters for each length is:
- For length 6: ( 36^6 )
- For length 7: ( 36^7 )
- For length 8: ( 36^8 )
Inclusion-Exclusion
Now, using the principle of inclusion-exclusion, we add back the number of passwords that miss two categories:
- Passwords with no digits and no uppercase letters (only lowercase letters): ( 26^6, 26^7, 26^8 )
- Passwords with no digits and no lowercase letters (only uppercase letters): ( 26^6, 26^7, 26^8 )
- Passwords with no uppercase and no lowercase letters (only digits): ( 10^6, 10^7, 10^8 )
Finally, subtract the number of passwords with no digits, no uppercase letters, and no lowercase letters (i.e., an empty password), which is zero.
Step 3: Calculation
Now, we sum up all possibilities for each password length (6, 7, 8) using inclusion-exclusion and subtract from the total number of possible passwords.
The total number of valid passwords is the sum of passwords for each length (6, 7, 8) minus the excluded invalid passwords.
Conclusion
This problem requires detailed computation using inclusion-exclusion to arrive at the exact number of valid passwords. You can apply this logic to get the precise total, but the main idea is to calculate the total number of passwords first and then subtract the invalid cases where the password doesn’t meet the criteria.