The correct answer is option 1 (Statements I and III) and here`s why:

Statement I says "All children are inquisitive", implying that every child is inquisitive. On the other hand, Statement III says "No children are inquisitive", which means that there`s not a single inquisitive child. These assertions directly contradict each other, therefore they cannot be true at the same time.

In contrast, it`s possible for both of these statements to be false at the same time. For example, if some children are inquisitive and others are not, both statements would be false.

Options 2, 3, and 4 contain pairs of statements that can be true simultaneously (for instance, it is possible for all children to be inquisitive and for some children to not be inquisitive, which would make both Statements I and IV true or false). Thus, the correct answer is option 1 (I and III).