I got 120 in two different ways.
The first method: (Over-counting then adjusting)
there are 6 faces and 6 colours so if we paint one at a time there are 6!=720 ways to paint it. However obviously some of these will be repeats. Because there are 6 different ways of coming up with the same result you divide by 6. So there are 720/6=120 ways.
The second method:
If we choose the first three colours and paint them all together, there is only one way of doing this. Or rather there are three identical ways. Then there are 3! ways of painting the other three faces in relation to the original three. But there were also 6C3 ways of choosing the original three colours. So the total no of ways= 3!x6C3 = 120 ways.
EDIT: There are actually 30 ways if you work it out systematically: MacMahon's Coloured Cubes
So my answers are 4x too big. Obviously the first time there were other ways of coming up with the same result that I didn't think of. Originally I thought it was just 6: One for each of the colours. But in fact there is 4 for each because I forgot that you can rotate each one 4 times. In the second method I forgot that you would end up with groups of 3 faces that formed between the first three and second three.
The first method: (Over-counting then adjusting)
there are 6 faces and 6 colours so if we paint one at a time there are 6!=720 ways to paint it. However obviously some of these will be repeats. Because there are 6 different ways of coming up with the same result you divide by 6. So there are 720/6=120 ways.
The second method:
If we choose the first three colours and paint them all together, there is only one way of doing this. Or rather there are three identical ways. Then there are 3! ways of painting the other three faces in relation to the original three. But there were also 6C3 ways of choosing the original three colours. So the total no of ways= 3!x6C3 = 120 ways.
EDIT: There are actually 30 ways if you work it out systematically: MacMahon's Coloured Cubes
So my answers are 4x too big. Obviously the first time there were other ways of coming up with the same result that I didn't think of. Originally I thought it was just 6: One for each of the colours. But in fact there is 4 for each because I forgot that you can rotate each one 4 times. In the second method I forgot that you would end up with groups of 3 faces that formed between the first three and second three.
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