How many small cubes will be formed having only one face colored?
Explanation:
Since a cube is cut into smaller cubes, the number of cuts hence pieces along each axis should be same.
If the number of pieces along each axis is p.
∴ p × p × p = 125
⇒ p = ∛125 = 5 pieces.
Below is the diagram representing these pieces.
When a bigger cube is cut into smaller cubes with n pieces along each axis.
⇒ Number of pieces with 1 side colored = 6(n – 2)2.
⇒ Number of pieces with 2 sides colored = 12(n – 2).
⇒ Number of pieces with 3 sides colored = 8.
⇒ Number of pieces with 0 side colored = (n – 2)3.
Here,
⇒ Number of pieces with exactly 1 side colored = 6(5 – 2)2 = 54.
⇒ Number of pieces with exactly 2 sides colored = 12(5 – 2) = 36.
⇒ Number of pieces with exactly 3 sides colored = 8.
⇒ Number of pieces with no side colored = (5 – 2)3 = 27.
Hence, option (a).
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