Step-by-Step Logic Solution:

1. Eliminate incorrect digits:

From Clue 2 [7 3 8], we know that "Nothing is correct." Therefore, the digits 7, 3, and 8 are not in the PIN.

2. Identify the first digit:

Look at Clue 5 [7 8 0]. Since 7 and 8 are eliminated, 0 must be a correct number. The clue says it is "wrongly placed" in the 3rd position. So, 0 must be in the 1st or 2nd position.

Now look at Clue 3 [9 0 6]. It says two numbers are correct but wrongly placed. We know 0 is one of them. Here, 0 is in the 2nd position and is "wrongly placed," which means 0 cannot be in the 2nd position. Combining this with the previous step, 0 must be in the 1st position.

3. Identify the third digit:

Look at Clue 1 [6 8 9]. We know 8 is not in the PIN. The clue says "One number is correct and correctly placed."
- If 6 were correct, it would be in the 1st position, but we already know 0 is in the 1st position.
- Therefore, 9 must be correct and correctly placed in the 3rd position.

4. Identify the middle digit:

From Step 2 and 3, we know 0 and 9 are the two correct numbers in Clue 3 [9 0 6]. This means 6 is not in the PIN.

Look at Clue 4 [6 1 5]. It says one number is correct but wrongly placed. Since 6 is out, either 1 or 5 is correct.
- If 1 were correct, it would be "wrongly placed" in the 2nd position, but that is the only spot left in our PIN [0 _ 9].
- Therefore, 5 must be the correct number. In Clue 4, it is in the 3rd position (wrongly placed), so it fits perfectly in the 2nd position.

Final Result:

The correct 3-digit PIN is 059.