How many sons does X have? Statement I: Q and U are brothers of T. Statement II: R is the sister of P and U. Statement III: R and T are daughters of X.

Introduction / Context:
This data sufficiency question involves several family members and asks for the number of sons of X. Three statements describe brothers, sisters and daughters. The challenge is to carefully interpret who is a child of X, which of those children are male, and whether that leads to a unique count of sons. Because there are three statements, we must consider combinations of them and finally decide whether even all three together are sufficient.

Given Data / Assumptions:

• We need the number of sons of X.
• Statement I: Q and U are brothers of T.
• Statement II: R is the sister of P and U.
• Statement III: R and T are daughters of X.
• Brother means male sibling and sister means female sibling in the usual sense.
• When someone is said to be brother or sister of another person, they share at least one common parent.
• We assume that when a child is described as daughter of X, that child is indeed a child of X.

Concept / Approach:
We must identify all children of X and then count which of them are sons. The statements tell us about gender and sibling relationships, but they do not always explicitly state which persons are children of X. Data sufficiency requires not only an answer, but a unique answer. If different consistent family structures give different numbers of sons, even when all statements are used, then the data are not sufficient.

Step-by-Step Solution:
Step 1: From statement III, R and T are daughters of X. This means R and T are female children of X. Step 2: From statement I, Q and U are brothers of T. Since T is a daughter of X, T is a child of X. Therefore, Q and U are siblings of T and hence also children of X. Being brothers, Q and U are male children of X, that is, sons of X. Step 3: So far, we know that X has at least four children: Q, U, R and T. Among these, Q and U are sons, and R and T are daughters. Step 4: From statement II, R is the sister of P and U. Since R is already known to be a daughter of X, R is a child of X. P and U are siblings of R, so P is also a child of X, and U is already known as a child of X. Step 5: However, the gender of P is not explicitly given. P is only described as a sibling of R and U, with R being a sister and U being a brother. Step 6: Thus, P could be male or female. If P is male, then P is an additional son of X. If P is female, then P is an additional daughter of X. Step 7: Therefore, the number of sons of X could be either two (Q and U) or three (Q, U and P), depending on whether P is female or male. Step 8: None of the statements specify the gender of P, and no logical deduction forces P to be male or female uniquely. Hence, even after using all three statements, the exact number of sons is not fixed.

Verification / Alternative check:
Scenario one: Assume P is male. Then children of X are Q, U, R, T and P. Among them, Q, U and P are sons, and R and T are daughters. Thus X has three sons. Scenario two: Assume P is female. Then children of X are Q, U, R, T and P. In this case, Q and U are sons, while R, T and P are daughters. Thus X has two sons. Both scenarios are perfectly consistent with all three statements: Q and U remain brothers of T, R remains sister of P and U, and R and T remain daughters of X. Therefore, the data do not determine a unique number of sons.

Why Other Options Are Wrong:
Options a, b and c claim sufficiency for different pairs of statements, but these pairs cannot determine the gender of P or even fully identify all children of X. Option d claims that all three statements together are sufficient. As shown, they still leave the gender of P undetermined, leading to uncertainty about the total number of sons. Hence only option e correctly recognises that the data are insufficient even when all statements are combined.

Common Pitfalls:
A common mistake is to assume that P is male simply because the other sibling mentioned with U is a sister, R. Some test takers also assume that the set of children consists of only the names that appear explicitly, without checking whether their genders are fixed. In data sufficiency problems involving family trees, it is crucial to track each person, whether they are a child of the key person and whether their gender is actually specified. If even one person can logically be either male or female, then the exact number of sons may remain ambiguous.

Final Answer:
Even when all three statements are taken together, the exact number of sons of X cannot be determined. Correct option: Even all the three statements I, II and III together are not sufficient to answer the question.