1. How many different tetrahedrons can be produced by coloring each face a solid color and using n different colors? (Two tetrahedrons are the same if they can be turned and placed side by side so that corresponding sides match in color.)

2. How many cubes with n colors?

Based on the corrected formulae:

  n  tetrahedron   cube
  1       1           1
  2       5          10
  3      15          57
  4      36         240
  5      75         800
  6     141        2226
  7     245        5390
  8     400       11712
  9     621       23355
 10     925       43450
 11    1331       76351
 12    1860      127920
 13    2535      205842
 14    3381      319970
 15    4425      482700
 16    5696      709376
 17    7225     1018725
 18    9045     1433322
 19   11191     1980085
 20   13700     2690800
 21   16611     3602676
 22   19965     4758930
 23   23805     6209402
 24   28176     8011200
 25   33125    10229375
 26   38701    12937626
 27   44955    16219035
 28   51940    20166832
 29   59711    24885190
 30   68325    30490050

using

   10   for N=1 to 30
   40     Tet=N+3*fnComb(N,2)+3*fnComb(N,3)+2*fnComb(N,4)
   50     Cub=N+8*fnComb(N,2)+30*fnComb(N,3)+68*fnComb(N,4)+75*fnComb(N,5)+30*fnComb(N,6)
   60     print using(3,0),N;using(8,0),Tet;using(12,0),Cub
   90   next
   99   end
  100   fnComb(N,R)
  110   local Ans
  120   if R>N then Ans=0:else Ans=combi(N,R)
  130   return(Ans)