combinatorics DS problem form MGMAT

[raja_asu]

A college admissions committee will grant a certain number of $10,000 scholarships, $5,000 scholarships, and $1,000 scholarships. If no student can receive more than one scholarship, how many different ways can the committee dole out 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.

The correct answer is C. which uses ANAGRAM method to solve the problem: 10!/(2!2!2!4!) and I understand that but for some reason I feel ANAGRAM method is not the right one for this kind of question.

My approach: One student can get only one of the 10K, 5K or 1K scholarships. If we have 6 scholarships doesn't matter how many 10K, 5K or 1K, the 1st scholarship can be taken by any of the 10 students : 10 ways.
2nd scholarship can be taken only by the remaining 9 students : 9 ways.
In this way, all the 6 scholarships can be distributed in : 10*9*8*7*6*5 ways.We don't need the no. of each type of scholarships as long each student gets only one of them.

In that case answer is A.

What am I missing here? Please clarify.

[StaceyKoprince]

Tricky one. You're not actually calculating in the correct way - start with the scholarships and match to students. Don't start with the students and match to scholarships.

Let's say I have five $10k scholarships and 1 $5k scholarship. The five $10k scholarships are identical - that is, there's no difference between getting the first $10k scholarship or the second one. There is, however a difference between getting the $10k scholarship and the $5k one.

I have six students named A, B, C, D, E, and F (each letter represents one, real, distinct person). Respectively, they get these scholarships: $10k, $10k, $10k, $10k, $10k, and $5k. (So, A through E get $10k and F gets $5k.) Let's say I now take the "first" and "second" $10k scholarships and swap them: A gets B's, and B gets A's. Is this a new way to dole out the scholarships? Nope, because the two are identical - A and B each still have a $10k scholarship. The only thing that would make a difference is if I swapped any one of my $10k students with F, who got only $5k. So, total, there's only 6 different ways for me to distribute these.

Now, let's say I have six $10k scholarships which go to A, B, C, D, E, and F. I can "swap" these around any way I want, but the result is the same: each student has a $10k scholarship. So there's only one way to distribute.

You can do an even more complicated example if you want: say you have three $10k, two $5k and one $1k. You have the same six students. Write down some possible combinations to show that you have more than one, and more than six, different ways to distribute these.
A, B, C, D, E, F
10,10,10,5,5,1
1,5,5,10,10,10
5,5,10,10,10,1
5,5,1,10,10,10
10,5,10,5,10,1
5,10,5,10, 10, 1
etc.