Answer
104,781,600
Work Step by Step
We simply choose the assigned number from the sets that are given. We know by the fundamental counting principle that we can multiply the independent outcomes together. $C(6,2)$ for freshmen, $C(8,3)$ for the sophomores, $C(12,4)$ for juniors, $C(10,5)$ for seniors.
Multiplying those numbers gives us $\boxed{104,781,600}$
