How many visitors saw the exhibition yesterday? Statement I: Each entry pass holder can take up to three persons along. Statement II: In all, 243 passes were sold yesterday.

Introduction / Context:
This data sufficiency problem is about counting visitors to an exhibition. You are asked to find how many visitors attended, given information about how many passes were sold and how many people each pass holder is allowed to bring. The challenge is to see whether the information is enough to determine a unique number of visitors, or whether multiple different totals remain possible.

Given Data / Assumptions:
- Question: How many visitors saw the exhibition yesterday?
- Statement I: Each entry pass holder can take up to three persons with him or her.
- Statement II: In all, 243 passes were sold yesterday.
- Every visitor enters through a pass holder, and there are no constraints on whether each pass holder must bring three persons or could bring fewer (including zero additional persons).

Concept / Approach:
We treat the total number of visitors as the sum of all pass holders plus all additional persons they bring. Statement I gives us the maximum number of people per pass but not the actual number for each pass holder. Statement II tells us how many passes were sold. For data sufficiency, if we cannot determine a unique total number of visitors because of flexible conditions, then the data are not sufficient.

Step-by-Step Solution:
Step 1: Analyze statement I alone. We know that each pass holder can take up to three persons along. So each pass could admit the pass holder plus anywhere from zero to three additional people, which means between 1 and 4 visitors per pass. But we do not know how many passes were sold, so statement I alone does not allow us to compute the total number of visitors. Step 2: Analyze statement II alone. We know that 243 passes were sold yesterday. However, we have no information about how many people each pass allows or how many people actually came with each pass holder. So statement II alone also does not provide enough information to determine the total number of visitors. Step 3: Combine statements I and II. With 243 passes sold and each pass holder allowed to bring up to three additional persons, the minimum number of visitors occurs when no pass holder brings anyone else. In that case, the number of visitors would be 243. Step 4: The maximum number of visitors occurs when every pass holder brings the full three additional persons. Then each pass admits 1 + 3 = 4 visitors, and the total number of visitors would be 243 * 4 = 972. Step 5: In between these extremes, any combination of pass holders bringing 0, 1, 2, or 3 additional persons can occur, so many different visitor totals between 243 and 972 are possible. Therefore, even with both statements together, we cannot determine a unique number of visitors.

Verification / Alternative check:
We can test specific cases. If half the pass holders bring one extra person and the other half bring none, the total visitor count will be some integer strictly between 243 and 972. There is nothing in the statements that forbids this scenario. If some bring two and some bring three, another total is possible. Because the problem allows up to three persons, and does not state that everyone always brings three, all these variations are consistent with the data, which proves that the total number of visitors is not fixed by the given information.

Why Other Options Are Wrong:
- Option a is wrong because statement I alone does not specify how many passes were sold, leaving the total completely open ended.
- Option b is wrong because statement II alone does not constrain how many visitors per pass there are.
- Option c is incorrect because neither statement alone is sufficient.
- Option d is wrong because even the combination of statements I and II allows a wide range of possible visitor counts; they do not pin down a single value.

Common Pitfalls:
Many students loosely assume that each pass holder actually brings three persons because that is the maximum allowed, but the wording "up to three" clearly includes scenarios where fewer are brought. Another error is to confuse the number of passes with the number of visitors. In data sufficiency questions, you must read constraints very literally and never assume that the maximum or minimum is always reached unless explicitly stated.

Final Answer:
Even when both statements are considered together, the exact number of visitors cannot be determined. Therefore, the correct data sufficiency choice is option E.