Explanation:

To find how many natural numbers up to 2001 are divisible by 3 or 4 but not 5:

1. Divisible by 3: There are 667 numbers divisible by 3 (3, 6, 9,..., 2001).

2. Divisible by 4: There are 500 numbers divisible by 4 (4, 8, 12,..., 2000).

3. Divisible by both 3 and 4 (or 12): There are 166 numbers divisible by 12 (12, 24, 36,..., 1992).

4. Divisible by 3 or 4: Using the inclusion-exclusion principle, \(667 + 500 - 166 = 1001\).

5. Divisible by 5: 400 numbers (5, 10, 15,..., 2000).

6. Divisible by 15: 133 numbers (15, 30, 45,..., 1995).

7. Divisible by 20: 100 numbers (20, 40, 60,..., 2000).

8. Divisible by 60: 33 numbers (60, 120, 180,..., 1980).

9. Divisible by 3 or 4, not 5: \(1001 - 400 + 133 + 100 - 33 = 801\).

- Option 1: Incorrect.

- Option 2: Correct, the total is 801.

- Option 3: Incorrect.

- Option 4: Incorrect.

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