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Combinatorics

by Winner2013 » Wed Oct 02, 2013 9:57 am
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.

Answer is c

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by theCodeToGMAT » Wed Oct 02, 2013 10:10 am
A = 10000
B = 5000
C = 1000

Statement 1:
Total distribution: 6
We don't have ratio of division.. maybe all these 6 were of 10000 or many more such cases
INSUFFICIENT

Statement 2:
A:B:C = 1:1:1
But, we dont know the total count
Maybe its 3 (1,1,1) or 6 (2,2,2) etc
INSUFFICIENT

Combining..
SUFFICIENT

Answer [spoiler]{C}[/spoiler]
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by GMATGuruNY » Wed Oct 02, 2013 10:41 am
Winner2013 wrote: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.
This is a PIECES-OF-THE-PUZZLE question.
Don't solve.
Determine whether there is SUFFICIENT information to solve.

Statement 1: In total, six scholarships will be granted.
A piece is missing: Are the scholarships all of the SAME type or of DIFFERENT types?
Without this information, we cannot determine the number of ways to distribute the scholarships.
INSUFFICIENT.

Statement 2: An equal number of scholarships will be granted at each scholarship level.
A piece is missing: How MANY of each type are being granted?
Without this information, we cannot determine the number of ways to distribute the scholarships.
INSUFFICIENT.

Statements 1 and 2 together:
Now we have ALL OF THE PIECES of the puzzle: two of each type of scholarship are being granted.
Thus, the total number of possible distributions can be determined.
SUFFICIENT.

The correct answer is C.
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