Find the total number of 4-digit numbers that do not contain the digit 3 or 6…

Concept: When you build a number place by place and each place is filled independently from an allowed set of digits, the total count is the product of the number of choices for each place (the multiplication principle). "Does not contain 3 or 6" means neither 3 nor 6 may appear in any place, so both digits are removed from the pool of allowed digits.

Application: A 4-digit number has four places: thousands, hundreds, tens, units.

1. Removing 3 and 6 from the ten digits 0-9 leaves 8 allowed digits: 0, 1, 2, 4, 5, 7, 8, 9.

2. Thousands place: it cannot be 0 (otherwise the number is not 4-digit), so the choices are 1, 2, 4, 5, 7, 8, 9 — that is 7 choices.

3. Hundreds, tens and units places: each may be any of the 8 allowed digits (0 is fine here), giving 8 choices for each of these 3 places.

4. Multiply the choices: 7 × 8 × 8 × 8 = 7 × 83 = 7 × 512 = 3584.

Cross-check: Counting all 4-digit numbers with no restriction gives 9 × 10 × 10 × 10 = 9000. Removing two of the ten digits keeps a fraction (8/10) of the choices at each free place and (7/9) at the leading place, so the restricted count stays well below 9000 and lands at 3584 — consistent with the direct product above.