If P $ Q means P is the brother of Q; P # Q means P is the mother of Q; P * Q means P is the daughter of Q in A # B $ C * D, who is the father?

The answer is D. Data is inadequate.

We can only determine the relationship between two people if we know both of their relationships to the same third person. In this case, we only know the relationships between A and B, and between C and D. We do not know the relationship between A and C, or between B and D. Therefore, we cannot determine who the father is.

For example, it is possible that A is the mother of B, and C is the daughter of D. In this case, A is not the father of C.

It is also possible that A is the brother of B, and C is the mother of D. In this case, A is not the father of C.

Without knowing the relationship between A and C, or between B and D, we cannot determine who the father is.