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. $21 \cdot 36^7$
b. $26: 36^7$
c. $26^5 \cdot 10^3$
d. $36^8
Answer (1)
The first character must be a letter, so there are 26 options.
The remaining 7 characters can be any of the 36 characters (letters or digits), giving 3 6 7 possibilities.
The total number of possible passwords is the product of these choices: 26 × 3 6 7 .
Therefore, the correct answer is 26 × 3 6 7 .
Explanation
Understand the problem 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.
Determine the number of possible characters There are 26 letters (A-Z) and 10 digits (0-9), so a total of 36 possible characters can be used to form the password.
Calculate the number of choices 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.
Calculate the total number of passwords The total number of passwords is the product of the number of choices for each position. This is calculated as 26 × 3 6 7 .
Select the correct option Comparing this to the given options, we see that option B, 26 × 3 6 7 , matches our calculation.
Examples
This type of problem is useful in cryptography and computer science when estimating the strength of passwords. For example, if a system requires an 8-character password that starts with a letter, this calculation helps determine how many possible passwords exist, which is crucial for assessing security.
