Problem Solving

Each year, a college admissions committee grants a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. The number of scholarships granted at each level does not vary from year to year, and no student can receive more than one scholarship. This year, how many different ways can the committee distribute the scholarships among the pool of 10 applicants?

(1) In total, six scholarships will be granted.

(2) An equal number of scholarships will be granted at each scholarship level.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

When looking at the information given in the question, one immediately notices that the number of scholarships to be awarded is not given, and without that number there is no way to determine how many different ways there are to award scholarships.

There could be ten scholarships, all a one level even, and so there would only be one way to distribute them to ten students.

Alternatively there could be one at each level, which situation would produce an answer entirely from the one produced by the first situation.

So at least we need to know how many scholarships there are to be awarded, and my initial inclination is to believe that we also need to know how many there are at each level.

Statement 1 tells us the number to be awarded. However, if the breakdown by level is one way, for instance, 6 at one level and none at the others, then there will be a certain number of ways to distribute them. If the breakdown by level is another way, maybe 2 of each, there will be a different number of ways to distribute the various types among 10 students.

So without even doing any math I am going to say that Statement 1 is insufficient. You can do math if you want, but I think that by considering different mixes of numbers at the levels and how they could be distributed, without actually calculating anything, you can figure out that different mixes of numbers adding up to 6 result in different numbers of ways that they can be distributed.

Statement 2 tells us that an equal number will be granted at each level, but those numbers could add up to any multiple of three that is less than 10. There could be 1 of each, or 3 total, there could be 2 of each, for a total of 6, or there could be 3 of each, for a total of 9 scholarships.

While it seems to me that there has to be a difference between how many ways 1 of each, 2 of each, and 3 of each can be distributed, I feel as if maybe I should confirm this by doing some math. Sometimes what seems to be the case is not the case, and in this case I feel as if I could get tricked.

One of each can be distributed in the following number of ways.

10C1 x 9C1 x 8C1 = 10 x 9 x 8 = 720

Two of each can be distributed in the following number of ways.

10C2 x 8C2 x 6C2 = 45 x 28 x something. I am not going to finish the math because already I can tell that this is going to be different from 720.

So the information given in Statement 2 can be used to produce multiple results, and so Statement 2 is insufficient.

Combined the statements tell us that we have 2 scholarships at each level, for a total of 6.

We already have the formula for this and there is a one number of unique ways these scholarships can be distributed.

So the correct answer is C.