Packing refers to the arrangement of constituent units in such a way that the forces of attraction among the constituent particles is the maximum and the contituents occupy the maximum available space. In two dimensions, there are hexagonal close packing and cubic close packing. In three dimentions, there are hexagonal, cubic as well as body centred close packings.
A certain oxide of a metal M crystallises in such a way that `O^(2-)` ions occupy hcp arrangement following ABAB... pattern. The metal ions however occupy `2//3` rd of the octahedral voids. The formula of the compound is :

Options

A. `M_(2)O_(3)`
B. `M_(3)O`
C. `M_(8//3)O_(3)`
D. `MO_(2)`

Explanation

To determine the formula of the compound formed by the metal M and oxide ions (O²⁻) based on the provided information, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure**:
- The oxide of metal M crystallizes in a hexagonal close-packed (hcp) arrangement.
- In hcp, the oxide ions (O²⁻) occupy the lattice points in an ABAB... pattern.

2. **Determining the Number of Ions**:
- In a hexagonal close-packed structure, there are 6 oxide ions per unit cell (n = 6).
- Therefore, the number of octahedral voids in the unit cell is also equal to the number of atoms per unit cell, which is n = 6.

3. **Metal Ion Occupation**:
- According to the problem, the metal ions occupy \( \frac{2}{3} \) of the octahedral voids.
- Since there are 6 octahedral voids, the number of octahedral voids occupied by metal ions is:
\[ \text{Number of metal ions} = \frac{2}{3} \times 6 = 4 \]

4. **Determining the Ratio**:
- We have 4 metal ions (M) and 6 oxide ions (O²⁻).
- The ratio of metal ions to oxide ions is:
\[ \text{Ratio} = 4 : 6 = \frac{2}{3} : 1 \]

5. **Converting to Whole Numbers**:
- To express the ratio in whole numbers, we can multiply both parts of the ratio by 3:
\[ 2 : 3 \]

6. **Writing the Formula**:
- From the ratio, we can write the empirical formula of the compound as:
\[ \text{Formula} = M_2O_3 \]

### Conclusion:
The formula of the compound formed by metal M and oxide ions is \( \text{M}_2\text{O}_3 \).