A cube having all of its sides painted is cut by two horizontal, two vertical, and other two planes so as to form 27 cubes all having the same dimensions. Of these cubes, a cube is selected at random.
The total number of cubes having at least one of its sides painted is

Options

A. (a)14
B. (b)18
C. (c)22
D. (d)26

Answer

To solve the problem, we need to determine the number of smaller cubes that have at least one of their sides painted after a larger cube has been painted and then cut into 27 smaller cubes.

### Step-by-Step Solution:

1. **Understanding the Structure**:
The original cube is painted on all sides and then cut into 27 smaller cubes. This means the original cube is divided into a 3x3x3 arrangement of smaller cubes.

2. **Total Number of Smaller Cubes**:
Since the cube is divided into 27 smaller cubes (3 cubes along each dimension), the total number of smaller cubes is:
\[ 3 \times 3 \times 3 = 27 \]

3. **Identifying Unpainted Cubes**:
The only smaller cubes that will not have any painted sides are those that are completely inside the larger cube. In a 3x3x3 cube, the only cube that is completely unpainted is the one in the center. This is because the center cube does not touch any of the outer faces of the larger cube.

4. **Counting Unpainted Cubes**:
There is only 1 cube in the center that has no sides painted.

5. **Calculating Painted Cubes**:
To find the number of cubes that have at least one side painted, we subtract the number of unpainted cubes from the total number of cubes:
\[ \text{Cubes with at least one side painted} = \text{Total cubes} - \text{Unpainted cubes} \]
\[ = 27 - 1 = 26 \]

6. **Final Answer**:
Therefore, the total number of cubes having at least one of its sides painted is 26.

### Conclusion:
The answer to the question is 26.

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