hey_thr67 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.
Let the number of $10,000 scholarships = X, the number of $5,000 scholarships = Y, and the number of $1,000 scholarships = Z.
Let T = Total number of scholarships awarded = X + Y + Z
3 types of scholarships will be awarded to 10 students, but we don't know how many of each type of scholarship will be awarded, and also we don't know how many students will get scholarships.
(1) X + Y + Z = T = 6, but we don't know X, Y, and Z individually, so there can be many possible combinations, viz., 3X, 2Y, 1Z or 4X, 1Y, 1Z. Since we don't get a unique answer, so statement 1 is NOT sufficient.
(2)An equal number of scholarships will be granted at each scholarship level implies it can be 1X, 1Y, 1Z or 2X, 2Y, 2Z or 3X, 3Y, 3Z, but again since we don't have a unique answer; NOT sufficient.
Combining (1) and (2), we have only one answer: 2X, 2Y, 2Z, which implies the combinations will be 10C2 * 8C2 * 6C2.
The correct answer is C. 
