Scholarships

Let's say:

x = number of $10,000 scholarships
y = number of $5,000 scholarships
z = number of $1,000 scholarships

no. of applicants = 10.

Statement 1: In total, six scholarships will be granted.

we have 10 applicants and 6 scholarships, so we can choose 6 people from the 10 applicants in 10C6 ways.

if all 6 scholarships are different, then we will have 6! ways to distribute within a certain combination of 6 recipients. however, in this case, we only have 3 different scholarships and we are unsure of x, y and z.

if x=1=y=1 and z=4, a given set of 6 recipients can receive scholarship in 6!/4 ways.
if x=y=z=2, a given set of 6 recipients can receive scholarship in 6!/(2x2x2) ways.

both these cases will yield different answers. hence insufficient.

Statement 2: An equal number of scholarships will be granted at each scholarship level.

this means x=y=z. however don't know the total number of scholarships awarded. insufficient.

both statements together:

Since we have a total of 6 scholarships, and the number of scholarships at each level is equal, we have x=y=z =2.

so for a given set of 6 recipients, we can distribute scholarships among them in 6!/(2x2x2) ways.

and we can choose 6 recipients from 10 applicants in 10C6 ways.

So the number of ways we can distribute scholarships among the pool of 10 applicants is (10C6) x (6!/8) ways. Sufficient.

Choose C.

-BM-