What is the total number of 4-digit numbers that do not contain the digit 3 or…

Concept: The Fundamental Principle of Counting states that if a task is completed in a sequence of independent stages, and stage 1 can be done in n₁ ways, stage 2 in n₂ ways, ..., stage k in nₖ ways, then the whole task can be completed in n₁ × n₂ × ... × nₖ ways. When a k-digit number is built digit-by-digit, each digit position is one stage, and the leading position additionally excludes 0 (otherwise the number would have fewer than k digits).

Application: Apply this to the 4 positions of the number, treating each as an independent stage.

1. Excluding the digits 3 and 6, the available digits are {0, 1, 2, 4, 5, 7, 8, 9} — 8 digits in total.

2. The first (thousands) position cannot be 0, since that would make the number fewer than 4 digits, so it also excludes 0 in addition to 3 and 6, leaving {1, 2, 4, 5, 7, 8, 9} — 7 valid digits.

3. Each of the remaining three positions (hundreds, tens, units) may be any of the 8 available digits, including 0.

4. By the Fundamental Principle of Counting, multiply the number of choices for each of the 4 positions: 7 × 8 × 8 × 8.

5. 7 × 8 = 56; 56 × 8 = 448; 448 × 8 = 3584.

Cross-check: Using complementary counting as an independent method: if 0 were also allowed in the leading position, all four positions could be filled from the 8 non-3/6 digits in 8 × 8 × 8 × 8 = 4096 ways. Among these, the cases where the leading digit is 0 fix the first position and leave the other three free, giving 1 × 8 × 8 × 8 = 512 ways. Subtracting, 4096 − 512 = 3584, confirming the direct computation.