Question

Of the 66 people in a certain auditorium, at most 6 people have their birthdays in any 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.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When solving problems involving distributions, look for constraints and relationships that directly affect the question being asked.

Answer 1

Answer: (E)

Step 1: Analyze statement (1).
Statement (1) tells us that more people have their birthday in February than in March, but this information does not provide any clue about January. Therefore, statement (1) alone is not sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that five people have their birthday in March, but this also does not give us any information about January. Therefore, statement (2) alone is not sufficient.
Step 3: Combine both statements.
Combining both statements, we know that:
- At most 6 people have their birthday in any given month.
- Five people have their birthday in March, so there is at most one person with a birthday in February or January.
- However, there is no guarantee that someone must have their birthday in January.
Therefore, the combined statements are still not sufficient.
\[ \boxed{E} \]