A group consist of 4 couples in which each of the 4 persons have one wife
Quantity I: Number of ways in which they could be arranged in a straight line such that the men and women occupy alternate positions
Quantity II: Eight times the number of ways in which they be seated around circular table such that men and women occupy alternate position.

Options

A. Quantity I ` gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge `Quantity II
D. Quantity I = Quantity II or No relation

Answer

To solve the problem step by step, we will find the values for Quantity I and Quantity II.

### Step 1: Calculate Quantity I
**Quantity I**: The number of ways to arrange 4 couples in a straight line such that men and women occupy alternate positions.

1. **Arrange the Men**:
- There are 4 men, and they can be arranged in \(4!\) ways.
\[ 4! = 24 \]

2. **Arrange the Women**:
- After arranging the men, we can arrange the 4 women in \(4!\) ways as well.
\[ 4! = 24 \]

3. **Total Arrangements**:
- Since the arrangement can start with either a man or a woman, we have two cases (starting with a man or starting with a woman).
- Therefore, the total arrangements will be:
\[ 2 \times (4! \times 4!) = 2 \times (24 \times 24) = 2 \times 576 = 1152 \]

### Step 2: Calculate Quantity II
**Quantity II**: Eight times the number of ways to seat the couples around a circular table such that men and women occupy alternate positions.

1. **Arrange the Men in a Circle**:
- When arranging in a circle, we fix one person to avoid counting rotations. Thus, the arrangement of 4 men can be done in \(3!\) ways.
\[ 3! = 6 \]

2. **Arrange the Women**:
- Once the men are seated, the 4 women can be arranged in \(4!\) ways.
\[ 4! = 24 \]

3. **Total Arrangements**:
- The total arrangements for seating around the circular table will be:
\[ 3! \times 4! = 6 \times 24 = 144 \]

4. **Multiply by 8**:
- Now, we multiply this by 8 to find Quantity II.
\[ 8 \times 144 = 1152 \]

### Conclusion
Now we compare the two quantities:
- **Quantity I**: 1152
- **Quantity II**: 1152

Thus, **Quantity I equals Quantity II**.

### Final Answer
Both quantities are equal, so the answer is:
**Option D: Quantity I equals Quantity II.**

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