In a cubic,`A` atoms are present on alternative corners, `B` atoms are present on alternate faces, and `C` atoms are present on alternalte edges and body centred of the cube. The simplest formula of the compound is

Options

A. `A_(2)BC_(4)`
B. `AB_(2)C_(4)`
C. `ABC_(4)`
D. `ABC_(2)`

Explanation

To find the simplest formula of the compound with the given arrangement of atoms in a cubic structure, we will analyze the contributions of each type of atom (A, B, and C) based on their positions in the cube.

### Step-by-Step Solution:

1. **Identify the Contribution of A Atoms:**
- A atoms are present at alternate corners of the cube.
- A cube has 8 corners, but since A atoms are at alternate corners, we only consider 4 corners.
- Each corner contributes \( \frac{1}{8} \) of an atom.
- Therefore, the total contribution from A atoms is:
\[ \text{Total A} = 4 \text{ corners} \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

2. **Identify the Contribution of B Atoms:**
- B atoms are present on alternate faces of the cube.
- A cube has 6 faces, but B atoms occupy alternate faces, which gives us 3 faces.
- Each face contributes \( \frac{1}{2} \) of an atom.
- Therefore, the total contribution from B atoms is:
\[ \text{Total B} = 3 \text{ faces} \times \frac{1}{2} = \frac{3}{2} \]

3. **Identify the Contribution of C Atoms:**
- C atoms are present on alternate edges and at the body center of the cube.
- A cube has 12 edges, and considering alternate edges gives us 6 edges.
- Each edge contributes \( \frac{1}{4} \) of an atom.
- The body center contributes 1 full atom.
- Therefore, the total contribution from C atoms is:
\[ \text{Total C} = 6 \text{ edges} \times \frac{1}{4} + 1 \text{ (body center)} = \frac{6}{4} + 1 = \frac{3}{2} + 1 = \frac{5}{2} \]

4. **Combine the Contributions:**
- Now we have:
- Total A = \( \frac{1}{2} \)
- Total B = \( \frac{3}{2} \)
- Total C = \( \frac{5}{2} \)

5. **Simplify the Formula:**
- To find the simplest formula, we can multiply all contributions by 2 to eliminate the fractions:
\[ A: \frac{1}{2} \times 2 = 1 \quad \Rightarrow A^1 \]
\[ B: \frac{3}{2} \times 2 = 3 \quad \Rightarrow B^3 \]
\[ C: \frac{5}{2} \times 2 = 5 \quad \Rightarrow C^5 \]
- Therefore, the simplest formula of the compound is:
\[ AB^3C^5 \]

### Final Answer:
The simplest formula of the compound is \( AB^3C^5 \).