Evaluating Statement I: Height Relationships

Statement (I) provides the following information about the heights:

• A is taller than B: A > B$
• B is shorter than A and E only. This implies:
  • B is shorter than A (A > B$)
  • B is shorter than E (E > B$)
  • Since B is shorter than *only* A and E among the group {A, B, C, D, E}, it means B is taller than the remaining individuals, C and D.
  • So, B is taller than C: B > C$
  • And B is taller than D: B > D$

From Statement (I), we know that A and E are both taller than B, and B is taller than C and D. This establishes that A and E are taller than B, C, and D. However, we do not know the height relationship between A and E. Therefore, we cannot determine how many people are taller than E based solely on Statement I.

Conclusion: Statement I alone is not sufficient.

Evaluating Statement II: Height Relationships

Statement (II) provides the following information:

• C is shorter than A: A > C$
• A is shorter than E: E > A$

Combining these, we get the relationship E > A > C$. This tells us that E is taller than A and C. However, we have no information about the heights of B and D relative to E. Therefore, we cannot determine how many people are taller than E based solely on Statement II.

Conclusion: Statement II alone is not sufficient.

Combining Statements I and II for Height Comparison

Now, let's combine the information from both statements:

• From Statement I: A > B$, E > B$, B > C$, B > D$.
• From Statement II: E > A$.

Let's consolidate the height orderings:

• We know E > A$ (from II).
• We know A > B$ (from I).
• We know B > C$ and B > D$ (from I).
• Also E > B$ (from I), which is consistent with E > A > B$.

This implies the complete ordering relative to E is:

• E is taller than A.
• E is taller than B (since E > A > B$).
• E is taller than C (since E > A > B > C$).
• E is taller than D (since E > A > B > D$).

Therefore, E is the tallest person in the group {A, B, C, D, E}. The question asks how many persons are taller than E. Since E is the tallest, no one is taller than E.

Conclusion: Both statements together are sufficient to answer the question.

Final Decision on Sufficiency

Since neither Statement I nor Statement II alone is sufficient to determine the number of people taller than E, but combining both statements allows us to definitively answer the question, we need both statements together.

Thus, the correct option is that both statement I and statement II together are required to answer the question.