https://file.aiquickdraw.com/mathbot/file/17305622391080ud9t3zc.jpg

To solve this problem, let's break down the requirements and analyze each step:

1. **Circular Arrangement**: Since the table is circular, fixing one person’s position eliminates rotational symmetry. Let's fix one person, say a male, in one position to simplify our calculations.

2. **Alternating Pattern**: With one male fixed, we have 7 other males and 8 females left. Since they are to alternate between male and female, once a male's position is set, the entire sequence of gender is determined around the table.

3. **Spouses Sitting Together**: Each male must sit directly next to their female spouse, meaning each male has only one female option next to him.

4. **Arrangements of Pairs**: Now we are essentially arranging 7 remaining couples in specific alternating spots around the fixed male.

### Steps to Solve:
1. **Arrange the Remaining Couples**: With one male fixed, we have 7 pairs left to arrange. Since each pair has a specific male-female combination, arranging these pairs around the table is equivalent to arranging the 7 pairs in a sequence.

   - There are \(7!\) ways to arrange the remaining 7 pairs around the table.

2. **Positioning of Individuals in Each Pair**: For each of these 7 pairs, we can either place the male or female first in the chosen position (since each pair is treated as a block in this arrangement). Therefore, for each pair, there are 2 ways to arrange the two people.

   - Thus, there are \(2^8\) ways to decide the individual arrangement within each pair (including the fixed pair).

### Total Calculation:
Combining these, the total number of ways is:
\[
7! \times 2^8
\]

#### Calculating Each Part:
1. \(7! = 5040\)
2. \(2^8 = 256\)

Multiplying these together gives:
\[
5040 \times 256 = 1290240
\]

### Conclusion:
The answer to the question is not provided in the options, suggesting there may be an error in the listed choices. However, based on the calculation above, the correct number of arrangements is **1,290,240**.

Would you like further clarification on any part of this solution?

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**Further Related Questions:**
1. How does fixing one person simplify the calculation in circular arrangements?
2. Why do we use \(2^8\) for the arrangements within each pair?
3. How would the calculation change if the table were not circular?
4. Could we use a different method to solve this problem?
5. What would be the total arrangements if there were 9 couples?

**Tip:** When dealing with circular permutations, always try fixing one person’s position to eliminate symmetry.

Eight male/female married couples are at a restaurant where they are to be seated at a circular table. In how many different ways can they be seated so that males and females alternate, and each person is seated by their spouse?

This problem explores the combinatorial challenge of arranging eight male/female married couples around a circular table, ensuring alternating genders and that each person sits beside their spouse. Learn how to calculate the total number of valid seating arrangements using circular permutations and alternating arrangements for paired elements.

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