Coloring Cubes
I have 5 paints : black , blue, red ,yellow, and green, and a cube. there are 2 side that will be painted with the same colored paint, and the other 4 side will be painted with the remaining paints (one paint each) .
How many ways are there to color the cube ?
Note : rotating the cube doesn't count.
The answer is 75.
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1 solution
We divide the problem into 2 cases :
the same colored paints are on the opposite side of the cube, then there are (4-1)! (cyclic permutation) way to color the cube, since the colored paints are on the opposite side, if we rotate the cube upside down , it will count as 2 , so we divide it by 2 , 3!/2!=3
the same colored paints are adjacent to each other, then there are 4! way to color the cube, again if we rotate the cube, (so that the the same colored paints position are swapped), coloring will count as 2, so we divide it by 2, 4!/2!=12
so the answer is (3+12) * ways to pick the same color , that is 15*5=75 .