A cube having all of its sides painted is cut to be 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.
If `P_3` is the probability that the cube selected has none of its sides painted, then the value of `27P_3` is:

Options

A. (a) `1`
B. (b) `2`
C. (c) `3`
D. (d) `5`

Answer

To solve the problem, we need to determine the probability \( P_3 \) that a randomly selected cube from the 27 smaller cubes has none of its sides painted.

### Step-by-Step Solution:

1. **Understand the Structure of the Cube**:
The original cube is painted on all sides and is then cut into 27 smaller cubes. This means the original cube is a \( 3 \times 3 \times 3 \) cube.

2. **Identify the Types of Smaller Cubes**:
- **Corner Cubes**: There are 8 corner cubes. Each of these has 3 faces painted.
- **Edge Cubes**: There are 12 edge cubes. Each of these has 2 faces painted.
- **Face Center Cubes**: There are 6 face center cubes. Each of these has 1 face painted.
- **Center Cube**: There is 1 center cube that has no faces painted.

3. **Count the Total Number of Cubes**:
The total number of smaller cubes is:
\[ 8 \text{ (corner)} + 12 \text{ (edge)} + 6 \text{ (face center)} + 1 \text{ (center)} = 27 \]

4. **Determine the Number of Unpainted Cubes**:
From the types of cubes identified, only the center cube has no sides painted. Therefore, the number of cubes with no sides painted is:
\[ 1 \text{ (center cube)} \]

5. **Calculate the Probability \( P_3 \)**:
The probability \( P_3 \) that a randomly selected cube has none of its sides painted is given by the ratio of the number of unpainted cubes to the total number of cubes:
\[ P_3 = \frac{\text{Number of unpainted cubes}}{\text{Total number of cubes}} = \frac{1}{27} \]

6. **Calculate \( 27P_3 \)**:
Now, we calculate \( 27P_3 \):
\[ 27P_3 = 27 \times \frac{1}{27} = 1 \]

### Final Answer:
Thus, the value of \( 27P_3 \) is \( 1 \).