Data Sufficiency — Family relations: How is M related to N? Statements: I. P has exactly two children, M and N, and is the mother-in-law of Q. Q is the sister-in-law of N. II. R, the sister-in-law of M, is the daughter-in-law of S, who has exactly two children, M and N.

Introduction / Context:
This problem involves kinship logic and the precise meaning of “sister-in-law,” “mother-in-law,” and “daughter-in-law.” We must determine the relationship of M to N (e.g., brother, sister) based on minimal, non-contradictory information, judging each statement’s sufficiency independently and in combination.

Given Data / Assumptions:

• “Sister-in-law” is the spouse of one’s sibling or the sister of one’s spouse, contextually resolved.
• “Mother-in-law” means parent of one’s spouse; “daughter-in-law” means the wife of one’s son.
• Each of P and S has exactly two children, M and N (as specified in the respective statements).

Concept / Approach:
Use Statement I to fix who is married to whom, then deduce genders and sibling relationships. For sufficiency, it is enough to identify how M is related to N (brother/sister) without fully mapping every relative in the family.

Step-by-Step Solution:
From I: P’s only children are M and N; P is mother-in-law of Q ⇒ Q is married to P’s child (either M or N).Given Q is the sister-in-law of N, Q must be married to N’s sibling (i.e., to M). Hence M is the son of P (because Q is a daughter-in-law), and Q is M’s wife.Therefore, M is a male sibling of N ⇒ M is N’s brother. This answers “How is M related to N?” uniquely.Statement II: R is sister-in-law of M and daughter-in-law of S; S’s only children are M and N. R must then be married to N (not M), which makes N the son of S. However, II does not reveal M’s gender. So II alone does not establish whether M is N’s brother or sister.

Verification / Alternative check:
Under I, no alternate consistent configuration makes Q sister-in-law of N without Q being M’s wife; thus M is male. Under II, multiple possibilities exist for M’s gender, preventing a definitive relationship label.

Why Other Options Are Wrong:

• II alone: Insufficient—M could be male or female.
• Either alone / Both: Only I alone is already sufficient; adding II is unnecessary.

Common Pitfalls:

• Misreading “sister-in-law” as necessarily spouse’s sister rather than the spouse of a sibling.
• Assuming equal information content of both statements; in fact, I pins down M’s gender.

Final Answer:
I alone is sufficient while II alone is not sufficient