Question

Students from four sections of a class accompanied by respective class teachers planned to go for a field trip. There were nineteen people in all. However, on the scheduled day one of the four teachers and a few students could not join the rest. Given below are some statements about the group of people who ultimately left for the trip.
I: Section A had the largest contingent.
II: Section B had fewer students than Section A.
III: Section C’s contingent was smaller than Section B.
IV: Section D had the smallest contingent.
V: The product of the number of student from each section is a multiple of 10.
VI: The number of students from Section C is more than 2.
VII: The product of the number of students from each section is a multiple of 24.
VIII: The largest contingent has more than 4 students.
IX: Each section contributed different number of students
The statements that taken together can give us the exact number of students from each section:

• A. I, II, III, IV, VI
• B. I, VI, VIII, IX
• C. I, II, III, IV, V, VI
• D. I, II, III, IV, VI, VII
• E. I, IV, VI, VII, IX

Hint:

In data sufficiency problems, the first step is to identify all variables. If a key variable (like the total number of items) is unknown and unconstrained, it is usually impossible to find a unique solution for the individual components.

Answer 1

Answer: (E)

Step 1: Identify the key missing information.
The initial plan involved 19 people: 4 teachers and 15 students (A+B+C+D = 15). On the day of the trip, 1 teacher and "a few students" were absent. The statements I through IX describe the group that *actually went*. The fundamental problem is that the total number of students who went on the trip (let's call it N) is not specified. The phrase "a few" is too vague to be a mathematical constraint.
Step 2: Analyze the nature of the problem.
To find the "exact number of students from each section," we need a set of constraints that leads to a single, unique solution for the numbers (A', B', C', D'). Without knowing the total N, it is highly unlikely that any combination of the given statements can constrain the problem enough to yield a unique answer.
Step 3: Test a set of constraints to demonstrate the ambiguity.
Let's test the statements from Option (D), which are very restrictive: I, II, III, IV, VI, VII.
From I, II, III, IV: We have the strict order A'>B'>C'>D'.
From VI: C' $\gt$ 2, which means C' 3.
This implies the minimum possible values are D'≥1, C'≥3, B'≥4, A'≥5.
From VII: The product A' × B' × C' × D' is a multiple of 24.
Let's search for solutions that fit these rules:
Possibility 1: Let the numbers be (5, 4, 3, 2). The order is correct. C' (3)>2. The product is 5×4×3×2 = 120. Since 120 / 24 = 5, this is a valid multiple. The total number of students would be N = 5+4+3+2 = 14. This is a plausible scenario (1 student absent).
Possibility 2: Let the numbers be (6, 4, 3, 2). The order is correct. C' (3)>2. The product is 6×4×3×2 = 144. Since 144 / 24 = 6, this is also a valid multiple. The total number of students would be N = 6+4+3+2 = 15. This is also a plausible scenario (0 students absent, which contradicts "a few students could not join" but may be mathematically possible).
Possibility 3: Let the numbers be (6, 5, 3, 1). The order is correct. C' (3)>2. The product is 6×5×3×1 = 90. This is not a multiple of 24.
Possibility 4: Let the numbers be (6, 5, 4, 1). The order is correct. C' (4)>2. The product is 6×5×4×1 = 120. This is a multiple of 24. Total N = 16 (not possible as it's more than the initial 15).
Since both (5, 4, 3, 2) and (6, 4, 3, 2) are potential solutions under the constraints of Option (D), this set of statements is not sufficient to find an "exact" number. The same logic applies to all other combinations of statements.
Step 4: Conclude on the question's validity.
The question is fundamentally flawed because the total number of students who participated is unknown. Without this crucial piece of information, no combination of the given statements can narrow the possibilities down to a single unique solution. Therefore, none of the options provided are sufficient. The question as stated is unsolvable. *(Note: While the provided answer key indicates (E), logical analysis shows that no option can be sufficient due to the ambiguity in the problem statement.)*
Therefore: Based on a rigorous analysis, none of the combinations are sufficient to find a unique solution, indicating a flaw in the question's premise. \[ \boxed{\text{None of the above combinations are sufficient.}} \]