The opposite faces of a 6 cm side cube is coloured with black, red and green and cut into 1 cm cubes then the number of cubes with only one face is coloured which is red is -

Correct option is C

Given:
A cube with a side length of 6 cm is colored on opposite faces with black, red, and green.

The cube is then cut into 1 cm cubes.

​Concept used:​

For a cube of side n × n × n painted on all sides which is uniformly cut into smaller cubes.

Solution:

When a 6 cm cube is cut into 1 cm cubes, the total number of smaller cubes will be: 6 × 6 × 6 = 216

Cubes with only one face colored will be located on the faces of the larger cube but not on the edges or corners. Each face of the cube has: 6 × 6 = 36

Each edge of a face has 6 cubes, and each face has 4 edges: 4 × 6 = 24

Cubes at the corners, which are included in the edges, must not be double-counted: 24 − 4 = 20 (since each face has 4 corners).

The number of cubes with only one face colored red (excluding the edges and corners) on a single face is,

36 − 20 = 16

Since there are only 2 faces colored red and opposite each other, the total number of cubes with only one face colored red is,

2 × 16 = 32

Thus, the correct answer is (c).

Alternative method:

The number of cubes with only one face is coloured which is red is =$$(n - 2)^2 \times 2$$​
​$$(6-2)^2 \times 2$$​
​$$16 \times 2 = 32$$​​