A cuboid of dimensions 4 in. X 6 in. X 7 in. is painted on all six fac

Question)
A cuboid of dimensions 4 in. X 6 in. X 7 in. is painted on all six faces with the same color. It is then completely cut into identical small cubes, each of side 1 in. What is the ratio of the number of cubes with no face painted to the number of cubes with exactly one face painted to those with exactly two faces painted?


Solution:
Size of the cuboid = 4 x 6 x 7

For cubes with no face colored:
We need to remove a layer of cubes from the top, bottom, left, right, front and back.
The resulting figure will have no color.
Thus, number of cubes = (4 - 2) x (6 - 2) x (7 - 2) = 40 cubes

For cubes with one face colored:
We need to find the central areas of each face (shown in gray in the image). Thus, there would be:
2 faces with (4 - 2) x (6 - 2) = 8 cubes each => 16 cubes
2 faces with (7 - 2) x (6 - 2) = 20 cubes each => 40 cubes
2 faces with (7 - 2) x (4 - 2) = 10 cubes each => 20 cubes
Thus, number of cubes = 16 + 40 + 20 = 76 cubes

For cubes with two faces colored:
We need to find the number of cubes along the edges (shown in brown in the image):
(Note: We should ignore the corners, since the corners are cubes with 3 faces colored)
There are 12 edges - 4 edges with 7 cubes along it, 4 edges with 6 cubes along it and 4 edges with 4 cubes along it. Thus, we have:
4 edges with (4 - 2) = 2 cubes each => 8 cubes
4 edges with (6 - 2) = 4 cubes each => 16 cubes
4 edges with (7 - 2) = 5 cubes each => 20 cubes
Thus, number of cubes = 8 + 16 + 20 = 44 cubes

=> Required ratio = 40 : 76 : 44 = 10 : 19 : 11

Answer A