Which of the following expressions shows the number of 8 character passwords that can be formed using letters and digits if the password must begin with a letter?
a. [tex]$21 \cdot 36^7$[/tex]
b. [tex]$26 \cdot 36^7$[/tex]
c. [tex]$26^5 \cdot 10^3$[/tex]
d. [tex]$36^8$[/tex]
Answer (1)
The first character must be a letter, so there are 26 options.
The next 7 characters can be any of the 36 (letters or digits), giving 3 6 7 possibilities.
Multiply these to find the total number of passwords.
The total number of possible passwords is 26 I m es 3 6 7 .
Explanation
Understand the problem and provided data We want to find the number of 8 character passwords that can be formed using letters and digits, with the condition that the password must begin with a letter. There are 26 letters (A-Z) and 10 digits (0-9), so a total of 36 possible characters (26 letters + 10 digits).
Calculate the possibilities for each position Since the first character must be a letter, there are 26 choices for the first character. The remaining 7 characters can be any of the 36 characters (letters or digits). Therefore, there are 3 6 7 choices for the remaining 7 characters.
Combine the possibilities The total number of passwords is the product of the number of choices for the first character and the number of choices for the remaining 7 characters, which is 26 I m es 3 6 7 .
State the final answer Therefore, the correct expression is 26 I m es 3 6 7 .
Examples
This type of problem is useful in cryptography and computer science when estimating the strength of passwords. For example, if a system requires 8-character passwords that start with a letter, this calculation helps determine how many different password combinations are possible, which is crucial for assessing security.
