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).

Answer 2

Instead of directly multiplying the choices for each digit, let's use complementary counting: count all 3-character strings from \( S=\{0,1,\dots,8\} \) and subtract those that don't qualify as genuine 3-digit numbers (i.e. those starting with 0).

With repetition allowed and 9 symbols available, the total number of 3-character strings (ignoring the leading-digit restriction) is \( 9 \times 9 \times 9 = 729 \). Among these, the strings that start with \( 0 \) are not valid 3-digit numbers; there are \( 1 \times 9 \times 9 = 81 \) such strings (first position fixed as 0, the other two positions free). Subtracting, the count of genuine 3-digit numbers is \( 729 - 81 = 648 \).

1. 900: This would be the count if all 10 digits (0 through 9) were available for each position without any leading-digit restriction; it doesn't match the restricted 9-symbol set \( S \) used here, so it is incorrect.
2. 720: This resembles a permutation count (like \( 9\times8\times7\)-style arrangements without repetition), not the repetition-allowed count required here, so it is incorrect.
3. 648: This matches \( 729 - 81 \) computed above via complementary counting.
4. None of these: Since \( 648 \) is in fact obtainable and is listed as option (C), this "none of these" option does not apply.

Complementary counting confirms the same total as the direct multiplication method.

Therefore, the correct answer is 648.