Each edge of a cube is equally divided into n parts , thus there are total ` n^(3)` smaller cubes . Let ,
`N_0 to ` Number of smaller cubes with no exposed surfaces
`N_1 to` Number of smaller cube with one exposed surface
`N_2` to Number of smaller cube with two exposed surface
`N_3 ` to Number of smaller cubes with three exposed surfaces
What is the value of `N_3 `?

Options

A. A)`(n-1)!`
B. B)`(n-2)^(2) `
C. C)`((n(n+1))/(2)`
D. D)`8`

Answer

To find the value of \( N_3 \), which represents the number of smaller cubes with three exposed surfaces when a cube is divided into \( n^3 \) smaller cubes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Cube Division**:
- A cube has 12 edges, and when each edge is divided into \( n \) equal parts, it creates \( n \) smaller cubes along each edge.
- Therefore, the total number of smaller cubes formed is \( n^3 \).

2. **Identifying Cubes with Three Exposed Surfaces**:
- The smaller cubes with three exposed surfaces are located at the corners of the cube.
- A cube has 8 corners.

3. **Conclusion**:
- Since each corner of the cube contributes one smaller cube with three exposed surfaces, the total number of smaller cubes with three exposed surfaces \( N_3 \) is equal to the number of corners in the cube.
- Therefore, \( N_3 = 8 \).

### Final Answer:
\[ N_3 = 8 \]

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