A permutations question, how to solve...?

A very difficult question to explain. The answer involves something called group theory and generating functions.

See the following link, which uses your cube problem as a basis for explanation.

u cant choose a face on a cube, coz we'll tell u they all just look the same to us.

it's easier to pick 2 opposite faces and made them UP and DOWN, so all the others just be side faces(4 of them).

but hold right there! we dont count the "picked" faces coz its really, really nothing.

now we have 6 colors right? we hv to pick 2 1st to color our UP-DOWN. that's

6C2=6!/4!2!=15

we picked, so now we paint. and mind u, it isnt like one side is up, and one down, coz if we rolled 2 faces away, in any direction, they'd just go upside down, wouldnt they?

now 4 more paints to go. theres only 2 ways about coloring the SIDE faces. why? because the connection among any 2 colors are; either they touch each other, or they dont. if 2 are facing each other, then u know the OTHER are also making an opposition. if 2 share a side, the OTHER 2 share another side. so, now we already told u why u need to times by 2.

2*6C2=2*15=30

I do not know. It seems as though the answer is 720.

side 1 can be chosen in 6 ways. Having chosen side 1, side 2 in 5 ways and so on.

6 x 5 x 4 x 3 x 2 x 1 = 720 ways.