In each of the following, one or more options ar correct. Choose the correct option(s). If `vec a, vec b, vec c` represent three concurrent edges of a rectangular parallelepiped whose lengths are 4,3,2 respectively then the value of `(vec a + vec b+ vec c)* (vec a xx vec b+vec bxx vec c+ vec c xx vec a)` is

Options

A. 0
B. 48
C. 72
D. none of these

Answer

To solve the problem, we need to evaluate the expression \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})\) given that the vectors represent the edges of a rectangular parallelepiped with lengths 4, 3, and 2.

### Step-by-step Solution:

1. **Define the vectors**:
- Let \(\vec{a} = 4\hat{i}\) (length 4 along x-axis)
- Let \(\vec{b} = 3\hat{j}\) (length 3 along y-axis)
- Let \(\vec{c} = 2\hat{k}\) (length 2 along z-axis)

2. **Calculate \(\vec{a} + \vec{b} + \vec{c}\)**:
\[ \vec{a} + \vec{b} + \vec{c} = 4\hat{i} + 3\hat{j} + 2\hat{k} \]

3. **Calculate the cross products**:
- \(\vec{a} \times \vec{b}\):
\[ \vec{a} \times \vec{b} = (4\hat{i}) \times (3\hat{j}) = 12\hat{k} \]
- \(\vec{b} \times \vec{c}\):
\[ \vec{b} \times \vec{c} = (3\hat{j}) \times (2\hat{k}) = 6\hat{i} \]
- \(\vec{c} \times \vec{a}\):
\[ \vec{c} \times \vec{a} = (2\hat{k}) \times (4\hat{i}) = -8\hat{j} \]

4. **Sum of the cross products**:
\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 12\hat{k} + 6\hat{i} - 8\hat{j} \]

5. **Combine the results**:
- Now we have:
\[ \vec{a} + \vec{b} + \vec{c} = 4\hat{i} + 3\hat{j} + 2\hat{k} \]
- And:
\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 6\hat{i} - 8\hat{j} + 12\hat{k} \]

6. **Calculate the dot product**:
\[ (4\hat{i} + 3\hat{j} + 2\hat{k}) \cdot (6\hat{i} - 8\hat{j} + 12\hat{k}) \]
- This expands to:
\[ 4 \cdot 6 + 3 \cdot (-8) + 2 \cdot 12 = 24 - 24 + 24 = 24 \]

7. **Final result**:
The value of \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})\) is \(24\).

### Conclusion:
The correct answer is not among the provided options (0, 48, 72, none of these). Therefore, the correct option is **none of these**.