varun007 wrote:You have a six-sided cube and six cans of paint, each a different color. You may not mix colors of paint. How many distinct ways can you paint the cube using a different color for each side? (If you can reorient a cube to look like another cube, then the two cubes are not distinct.)

(A) 24
(B) 30
(C) 48
(D) 60
(E) 120

pls can somebody explain in detail

That is a tricky problem. Say we have six colours, Red, Green, Yellow, Purple, Black and White.

I must paint one side Red. I then have 5 choices for the colour which is opposite Red. Let's pretend that Purple is opposite Red for now, and we'll multiply by 5 at the end to get the total (and notice we're already down to B, D or E, because the answer must be divisible by 5).

Now, one of the four remaining faces must be Yellow. There are 3 choices for which colour should be opposite Yellow, then 2 choices for how to arrange the remaining two colours.

So there should be 5*3*2 = 30 choices in total.