M is the son of P, N is the sister of M, O is the father of N. How is P…

In blood-relation problems, siblings such as brothers and sisters share the same parents. Within a single nuclear family, only one person occupies the role of father and only one the role of mother -- so once one parental role for a child is already assigned to a specific person, the remaining parent must be the other person in the family.

1. M is the son of P, so P is a parent of M.

2. N is the sister of M, so N and M share the same parents -- N is also a child of P.

3. O is the father of N, so the father's role for N is already held by O.

4. Since P is a parent of N but cannot also be the father (that role belongs to O), P must be the mother of N.

Cross-check with the family tree: the parents are P (mother) and O (father), and their children are M (son) and N (daughter, sister of M) -- every relation in the statement holds without contradiction.