Introduction:
In a circular table seating arrangement, the seats are arranged in a circular shape. The number of ways to seat a party of 4 men and 4 women at a circular table can be determined using the concept of circular permutations.

Explanation:
To solve this problem, we can follow the steps below:

Step 1: Fix a reference point:
To eliminate overcounting, we fix a reference point at the table. We can imagine the table as a clock with the reference point as the 12 o'clock position.

Step 2: Arrange the men:
Since there are 4 men, they can be arranged in a circular manner in (4-1)! = 3! = 6 ways around the table. This is because once the reference point is fixed, the other 3 men can be arranged in a linear manner in 3! ways.

Step 3: Arrange the women:
Similarly, the 4 women can be arranged in a circular manner in (4-1)! = 3! = 6 ways around the table.

Step 4: Calculate the total number of arrangements:
To calculate the total number of arrangements, we multiply the number of arrangements of men and women together since these arrangements are independent of each other.

Total number of arrangements = 6 (men) * 6 (women) = 36

Conclusion:
Therefore, there are 36 different ways to seat a party of 4 men and 4 women at a circular table.