A cuboid of dimensions 7 cm × 5 cm × 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm × 5 cm, 5 cm × 3 cm, 7 cm × 3 cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1 cm. Which of the following statements is/are correct ?
1. There are exactly 15 small cubes with no paint on any face.
2. There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.

Select the correct answer using the code given below :

Correct Option: 1 only

A cuboid of dimensions 7 cm × 5 cm × 3 cm is cut into 1 cm small cubes. Total small cubes = 7 × 5 × 3 = 105. Let us evaluate statement 1: 'There are exactly 15 small cubes with no paint on any face.' Internal cubes with no paint have dimensions (7 - 2) × (5 - 2) × (3 - 2) = 5 × 3 × 1 = 15 cubes. Thus, statement 1 is correct. Let us evaluate statement 2: 'There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.' Two-painted cubes lie along the edges of the cuboid. The dimensions are length l = 7, breadth b = 5, height h = 3. Opposite faces 7×5 are red, 5×3 are green, 7×3 are blue. Edges formed by the intersection of green faces (5×3) and blue faces (7×3) are parallel to the length (7 cm). There are 4 such edges of length 7 cm. Excluding the 2 corner cubes on each such edge (which have 3 painted faces), each such edge has (7 - 2) = 5 inner cubes with 2 painted faces (green and blue). Since there are 4 such edges, the total number of cubes with green and blue faces is 4 × 5 = 20 cubes (not 6). Thus, statement 2 is incorrect. Therefore, only statement 1 is correct.