This question is a logical reasoning puzzle involving contradicting statements.

Option 1: Statements I and IV - Statement I says all balls are black, and IV says no balls are black. These statements can`t both be true at the same time (since they contradict each other), but they could both be false (if some, but not all, balls are black). So, this is the correct answer.

Option 2: Statements I and III - These two can both be true (all balls are black and only some balls are black - they both could be true if `some` is interpreted as `all`). But they also both could be false at the same time (if no balls are black or all balls are white).

Option 3: Statements II and III - They can both be true (all balls are white and only some balls are black - if `some` is understood as `none`). They also could be both false (if all balls are black).

Option 4: Statements III and IV - They can be true both at the same time (only some balls are black and no balls are black - if `some` is interpreted as `none`). But they also could both be false at the same time (if all balls are black).

So