May 2005
Answer:
Part 1:
6 * 68C3 * 122C3 = 6 * 50116 * 295240 = 88777487040.
Select three of the 68 rows (in 68C3=50116 possible ways), and three of the 122 columns (in 122C3=295240 ways); in the resulting 3x3 pattern, there are six ways to arrange 6 empty compartments and 3 black marbles with one marble in each row and each column. All other compartments receive black marbles.
Part 2:
0.
Each "occupied" plane (a plane with nonzero number of clear cubes) has to have at least four clear cubes. So there can't be two horizontal occupied planes (parallel to x-y plane). So some vertical line has an odd number of clear cubes, hence an odd number of red cubes.
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