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There are N tetrahedra of one color each.
If there are two colors, there are C(N,2) ways of choosing the two colors, and then either they each appear on two of the faces or one or the other is in the majority 3-to-1, making 3*C(N,2) in all.
If there are three colors, there are C(N,3) ways of choosing the three colors and any one of the three can be the duplicated color. Once the tetrahedron is colored, there's no difference in handedness, as the tetrahedron can be flipped over interchanging the non-identical faces. So there are 3*C(N,3) in all.
If there are four colors, there are C(N,4) ways of choosing the four colors and this time mirror reversing does create a different tetrahedron, and so there are 2*C(N,4) in all.
Tabulated:
N 2-color 3-color 4-color
4 18 12 2
5 30 30 10
6 45 60 30
7 63 105 70
8 84 168 140
9 108 252 252
10 135 360 420
11 165 495 660
12 198 660 990
13 234 858 1430
14 273 1092 2002
15 315 1365 2730
16 360 1680 3640
17 408 2040 4760
18 459 2448 6120
19 513 2907 7752
20 570 3420 9690
N = 9 is the only one where the number of 3-color tetrahedra matches the number of 4-color tetrahedra, and so there are 9 cans of paint to choose from. There are 9 + 108 + 252 + 252 = 621 tetrahedra in all.
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