Question

How many different 5-character passwords can be formed using the digits 0-9 and lowercase letters a-z if the first character must be a letter and repetitions are allowed?

• A. \( 36^5 \)
• B. \( 26 \times 35^4 \)
• C. \( 26 \times 10 \times 36^3 \)

Hint:

When repetitions are allowed, the number of choices remains constant for each position. If repetitions were NOT allowed, the number of choices would decrease by one for each subsequent position.

Answer 1

The Correct Option is A

Concept: The Fundamental Counting Principle (Product Rule) states that if there are \(n\) ways to do one thing, and \(m\) ways to do another, then there are \(n \times m\) ways to do both.
• Total digits (0-9) = 10
• Total lowercase letters (a-z) = 26
• Total available characters = \(10 + 26 = 36\)

Step 1: Determining the choices for the first character.
The problem states that the first character must be a letter. Since there are 26 lowercase letters available: \[ \text{Choices for Position 1} = 26 \]

Step 2: Determining the choices for the remaining four characters.
For the next 4 positions, any character (digit or letter) can be used, and repetitions are allowed. Each position has 36 possible choices: \[ \text{Choices for Position 2, 3, 4, and 5} = 36 \times 36 \times 36 \times 36 = 36^4 \]

Step 3: Applying the product rule.
Total number of passwords = (Choices for Pos 1) \(\times\) (Choices for remaining positions): \[ \text{Total} = 26 \times 36^4 \]