haha, misread one info. The revised answer -->

I thouht they have all th3 3 students in common...

Here is the revised answer:

1) Teacher A:
- picks 1 (common) student out of 10 = C(10, 1)
- picks a rank (1 out of 3) for this student = P(3, 1)
- picks another 2 out of 9 and ranks them = P(9, 2)

2) Teacher B:
- picks a rank for the 1st student, 1 out of 2, = P(2, 1)
- picks another 2 out of 7 and ranks them = P(7, 2)

3) Teacher C:
- picks 2 out of 5 and ranks them = P(5, 2)

then ... it actually = P(10) :P