Question

The number of 3 digit numbers that can be formed with \( S = \{0,1,2,3,\dots,8\} \) when repetitions are allowed is

• A. 900
• B. 720
• C. 648
• D. None of these

Hint:

When forming numbers with repetition allowed, consider the number of possible choices for each digit and multiply them together.

Answer 1

The Correct Option is C

Step 1: Understand the number formation.
We need to form 3-digit numbers using the digits from \( S = \{0,1,2,3,\dots,8\} \). The number must have three digits, and repetition is allowed.
Step 2: Analyze the options.
- The first digit of the 3-digit number cannot be 0, so it can be any one of the digits from \( \{1, 2, 3, \dots, 8\} \). This gives us 8 possible choices for the first digit.
- The second and third digits can each be any of the 9 digits from \( \{0, 1, 2, 3, \dots, 8\} \), which gives us 9 possible choices for each of these digits.

Step 3: Calculate the total number of numbers.
The total number of 3-digit numbers is the product of the choices for each digit: \[ 8 \times 9 \times 9 = 648. \]
Step 4: Conclusion.
Thus, the number of 3-digit numbers that can be formed is \( 648 \), which corresponds to option (C).