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The GMAT is scored on a scale of \(200\) to \(800\) in \(10\) point increments. (Thus \(410\) and \(760\) are real GMAT scores but \(412\) and \(765\) are not). A first-year class at a certain business school consists of \(478\) students. Did any students of the same gender in the first-year class who were born in the same-named month have the same GMAT score?
We need to determine whether any students of the same gender born in the same month have the same GMAT score.
This is a yes/no question. We need to establish whether we can definitively answer:
Since there are \(2\) genders and \(12\) months, we have \(24\) different gender-month combinations. This is a classic setup for the Pigeonhole Principle: if we can show that any gender-month group has more students than available GMAT scores, then some students in that group must share the same score.
Statement 1 tells us that GMAT scores in the class range from \(600\) to \(780\).
This dramatically reduces our available GMAT scores:
With \(478\) students distributed across \(24\) gender-month combinations:
Here's the crucial realization: we have an average of about 20 students per group, but only 19 possible GMAT scores.
By the Pigeonhole Principle, at least one gender-month group must have 20 or more students. Since there are only 19 possible scores available for that group, at least two students in that group must have the same GMAT score.
Statement 1 is sufficient to answer the question. The answer is definitively YES.
[STOP - Statement 1 is Sufficient!]
This eliminates choices B, C, and E. We only need to check if Statement 2 alone is also sufficient.
Important: We now forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us that \(60%\) of the students are male.
Now we have all \(61\) possible GMAT scores available (\(200\) to \(800\)), and:
Even with an average of 24 males per month, we have 61 possible GMAT scores - more than enough to give each male born in any given month a different score. The same logic applies to females.
Without knowing the actual distribution or score range, we cannot definitively conclude that duplicates must exist.
Statement 2 is NOT sufficient to answer the question.
This eliminates choices B and D.
Since only Statement 1 is sufficient to determine that some students of the same gender born in the same month must have the same GMAT score, the answer is A.
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."