Select the close-packing arrangements in the following:
a. … `ABABABA` …, b. … `ABCABCABCA` …
c. … `ABABCBCABC` … , d. … `AC CBCABCABC` …

Explanation

To determine the close-packing arrangements from the given options, we will analyze each arrangement step by step.

### Step-by-Step Solution:

1. **Option A: `ABABABA`**
- This arrangement alternates between A and B.
- When visualized in a 2D layer, it can be represented as:
```
A B A
B A B
A B A
```
- This structure is a type of close packing known as **CCP (Cubic Close Packing)** or **HCP (Hexagonal Close Packing)**, as it allows for efficient packing of spheres.
- **Conclusion**: This is a close-packing arrangement.

2. **Option B: `ABCABCABC`**
- This arrangement has three different types of spheres (A, B, and C) repeating in a sequence.
- When visualized, it can be represented as:
```
A B C
A B C
A B C
```
- This arrangement corresponds to **HCP (Hexagonal Close Packing)**, where each layer is offset from the previous one.
- **Conclusion**: This is a close-packing arrangement.

3. **Option C: `ABABCBCABC`**
- This arrangement is a bit more complex as it mixes A, B, and C in a non-repetitive manner.
- Visualizing this arrangement:
```
A B A
B C B
A B C
```
- This structure can be seen as a combination of CCP and HCP arrangements, allowing for efficient packing.
- **Conclusion**: This is a close-packing arrangement.

4. **Option D: `AC CBCABCABC`**
- This arrangement does not follow a regular pattern and appears more random.
- When visualized, it does not maintain a consistent layer structure:
```
A C
C B
C A
B C
A B
C
```
- This arrangement does not represent a close packing structure as it lacks the regularity required for efficient packing.
- **Conclusion**: This is not a close-packing arrangement.

### Final Answer:
The close-packing arrangements are:
- **Option A: `ABABABA`**
- **Option B: `ABCABCABC`**
- **Option C: `ABABCBCABC`**

**Option D is not a close-packing arrangement.**

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