Later, each was asked "How many other students were in your room?" The average of these responses was 12.

How is this possible?

12=1/37 * [(x*(x-1)+y*(y-1)+z*(z-1)]

Simplifies to:

X^2+Y^2+Z^2=37*13=481

Since 481=400+81=256+144+81

16,  12,  9  is the triplet defining the distribution of the students,

Piece of cake!