Question

Of the 66 people in a certain auditorium, at most 6 people have their birthdays in any one given month. Does at least one person in the auditorium have a birthday in January?
(1) More of the people in the auditorium have their birthday in February than in March.
(2) Five of the people in the auditorium have their birthday in March.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

For "Yes/No" questions, especially in logic problems, a powerful strategy is to assume the opposite of what the question asks for and look for a contradiction. Here, assuming "No" (zero birthdays in January) leads to a very specific scenario that is contradicted by both statements.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a logic and combinatorics problem that can be approached using the Pigeonhole Principle or by testing extreme scenarios.
Total people = 66.
Number of months = 12.
Maximum people per month \(\le\) 6.
The question is: Is the number of people with a birthday in January \(\ge 1\)? This is a "Yes/No" question.
To prove sufficiency, we must show that the number for January cannot be 0. Let's test the "No" case: assume there are 0 people with a birthday in January.
Step 2: Detailed Explanation:
If there are 0 birthdays in January, then all 66 people must have birthdays in the remaining 11 months.
The maximum number of people that can be accommodated in these 11 months is \(11 \text{ months} \times 6 \text{ people/month} = 66 \text{ people}\).
This means that for the "No" case (0 birthdays in January) to be possible, every single one of the other 11 months must have exactly 6 birthdays. So, the "No" case requires: Jan=0, Feb=6, Mar=6, Apr=6, ..., Dec=6.
Analyze Statement (1): More of the people in the auditorium have their birthday in February than in March.
This means the number of birthdays in February>the number of birthdays in March.
The only scenario that gives a "No" answer to the main question requires Feb=6 and Mar=6. This contradicts the statement that Feb>Mar.
Therefore, the "No" scenario is impossible under this condition. The number of birthdays in January must be greater than 0. The answer is definitively "Yes".
Statement (1) is sufficient.
Analyze Statement (2): Five of the people in the auditorium have their birthday in March.
This means the number of birthdays in March = 5.
The only scenario that gives a "No" answer to the main question requires Mar=6. This contradicts the statement that Mar=5.
Therefore, the "No" scenario is impossible under this condition. The number of birthdays in January must be greater than 0. The answer is definitively "Yes".
Statement (2) is sufficient.
Step 3: Final Answer:
Each statement alone is sufficient to answer the question.