Permutation and Combinations

That's good Question. I doubt whether we can get these kind of high difficulty level question in GMAT at present.

My Solution:

First of all don't get confused by here six-sided cube means six faced cube.

You have to imagine the picture of cube in mind as I can't draw cube here.

Take one face of the cube and paint it with any 1 of the 6 colors mark it as Top of cube.

Now take the face parallel to this painted face and paint it with one of the remaining 5 colors.


So now we get a cube whose 2 parallel faces are painted and 4 adjacent faces are unpainted.

these unpainted faces form a closed figure and can be treated as circle for the P&C application. (Just imagine the figure in mind or Take a Rubik' cube and see it by applying above steps.)

NOTE: We are painting 2 parallel faces as it will create a closed figure of unpainted faces.

Now, we have to paint 4 faced closed figure and that can be done in (4-1)! = 3! because we have circular permutation.


Reasoning for circular permutation- The difference between linear and circular arrangement is that in linear we have a beginning and an end but in circular we have the place for beginning and end or we can say 1 place is always fixed.

Apply the same in 4 faced close figure. 1 face is already fixed after you painted it with 1 0f 4 colors.


Hope it helps!


Thanks & Regards,
Vishal

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

Total number of ways to paint the cube = (total number of ways to arrange the 6 colors)/(total number of ways to orient the cube).

Total number of ways to arrange the 6 colors = 6! = 720.
Total number of ways to orient the cube = 24.
Total number of ways to paint the cube = 720/24 = 30.

The correct answer is B.

Here is one way to count the number of ways that each arrangement can be oriented:
Let say that the 6 colors are A, B, C, D, E and F and that the cube has been painted so that A and B are on opposite faces.
The cube can be oriented so that any of the 6 colors is on top.
Number of options for the color on the top face = 6.
Let's say that the cube has been oriented so that A is on top.
Since A and B are on opposite faces, B is on the bottom.
The cube can now be rotated clockwise so that either C, D, E or F is facing forward.
Number of options for the forward face = 4.
To combine the number of options for the top face and the number of options for the forward face, we multiply:
6*4 = 24.