There must be at least 3 questions with scores of 3, for if there
were only 2 with scores of 3, that would be only be 6, leaving the
other 15-6 or 9 to be made with 3 scores of 2 or less, which is
impossible.  

So the 3 scores of 3 each accounts for 9, so the other 15-9 or 6,
must be made with 3 questions.  The only ways to get 6 with
3 questions is 0+3+3, 1+2+3, or 2+2+2.

0+3+3+3+3+3 = 15
1+2+3+3+3+3 = 15
2+2+2+3+3+3 = 15

So there are only 3 basic ways to get 15.  However we must permute these
over all 6 questions.

The number of distinct permutations of 0,3,3,3,3,3 is 
The number of distinct permutations of 1,2,3,3,3,3 is 
The number of distinct permutations of 2,2,2,3,3,3 is 

Answer: 6+30+20 = 56 ways possible to get 15  

Edwin