How many sons does D have? Statement I: The father of A has three children. Statement II: B is the brother of A and B is a son of D.

Introduction / Context:
This data sufficiency question focuses on family relationships and counts of children. We are asked to determine how many sons a person D has. Two statements are provided involving A, B, and the father of A. The challenge is to understand what can and cannot be concluded from these pieces of information, and to recognise when the information is not enough to determine a unique number of sons.

Given Data / Assumptions:

• We need the number of sons of D.
• Statement I: The father of A has three children.
• Statement II: B is the brother of A and B is a son of D.
• Brother means a male sibling in the usual sense.
• No explicit statement connects the father of A in statement I to D in statement II.
• All children mentioned are living, and there may be additional children not mentioned.

Concept / Approach:
The method in data sufficiency is to evaluate:

• Statement I alone.
• Statement II alone.
• Both statements together.

If there exists more than one possible family structure consistent with the statements but with different numbers of sons for D, then the data are not sufficient. We must avoid silently adding assumptions such as equating the father of A with D unless it is clearly stated.

Step-by-Step Solution:
Step 1: From statement I, the father of A has three children. Let that father be F. Then F has exactly three children, one of whom is A. The other two children are not identified by name or gender. Step 2: Statement I alone does not even mention D, so it clearly cannot tell us how many sons D has. Therefore statement I alone is not sufficient. Step 3: From statement II, B is the brother of A and is a son of D. Thus B is a male child of D, and B is a sibling of A. Step 4: From statement II alone, D has at least one son, namely B. However, D may have other sons and daughters not mentioned. Hence we cannot determine the exact number of sons of D from statement II alone. Step 5: Now combine both statements. One possible interpretation is that F, the father of A in statement I, might be the same person as D. But this is not explicitly stated. Even if we assume that D is the father of A, we only know that D has three children in total. Step 6: Among these three children, we know from statement II that B is a son and sibling of A. However, the genders of A and the third child are not specified. Step 7: Different configurations are possible. For example, D could have one son B and two daughters including A, giving one son. Alternatively, D could have two sons, B and the third child, with A as a daughter, giving two sons. It is also possible that A is also a son, giving three sons. Step 8: Since multiple consistent family patterns produce different numbers of sons for D, even the combined information does not uniquely determine the required number.

Verification / Alternative check:
Construct scenario one: Let D be the father of A. Suppose A is female, B is male and the third child is female. Then D has one son. Construct scenario two: Again let D be the father of A. Suppose A is female, B is male and the third child is male. Then D has two sons. Construct scenario three: Suppose both A and B are male and the third child is female. Then D has two sons again. In every case, the given statements I and II remain valid, but the number of sons changes or at least is not fixed to one unique value.

Why Other Options Are Wrong:
Option a is wrong because statement I does not mention D at all. Option b is wrong because statement II only guarantees at least one son for D, not the exact count. Option c is wrong because neither statement alone gives a unique answer. Option d is wrong because even after combining the statements, several different distributions of sons and daughters are possible.

Common Pitfalls:
A common mistake is to assume that the father of A mentioned in statement I must be D. Even if we make that assumption, students often conclude that D has three children and then wrongly infer that the number of sons is fixed, sometimes assuming an equal split between sons and daughters without any basis in the statements. Data sufficiency questions require strict logical discipline. Only what is explicitly given or is a direct consequence of what is given should be used. Any additional assumptions about genders or identities of parents can lead to incorrect conclusions about sufficiency.

Final Answer:
The exact number of sons of D cannot be determined from the given statements. Correct option: Even using both statements I and II together, the data are not sufficient to answer the question.