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: Total 3-character strings from the 9 symbols, repetition allowed: \(9\times9\times9=729\).
Step 2: Subtract strings starting with 0 (not valid 3-digit numbers): \(1\times9\times9=81\).
\[ \boxed{729 - 81 = 648} \]

Answer 2

A third way is to directly count using the multiplication principle but grouping the first digit's choices explicitly by cases (odd digit, even non-zero digit, etc. is unnecessary complexity here) — more simply, treat the three positions as independent slots and multiply the counts, cross-checking against the complement method already used.

Position 1 (hundreds): must be nonzero, so it can be any of \( \{1,2,\dots,8\} \) — 8 choices. Position 2 (tens): any of the 9 symbols in \( S \), since leading-zero restriction does not apply here — 9 choices. Position 3 (units): similarly 9 choices.

1. 900: This would follow from \( 9 \times 10 \times 10 \) or similar, i.e. treating the digit set as having 10 symbols; but \( S \) only has 9 symbols, so this option is incorrect.
2. 720: This would arise from a without-repetition arrangement count, not matching the "repetitions are allowed" condition of the problem, so it is incorrect.
3. 648: This equals \( 8 \times 9 \times 9 \), matching the slot-by-slot count above, and also matches the complementary-counting total.
4. None of these: Not applicable, since \( 648 \) is exactly obtained and listed as an option.

Both the direct slot count and the complementary count give the same total.

Therefore, the correct answer is 648.