To solve this problem, let's break it down step by step:

Step 1: Understanding the problem

• We have 6 chairs arranged around a circular table, and we need to seat 3 men and 3 women such that no two men are opposite each other.
• "Opposite chairs" means chairs that are directly across from one another on the circle.

Step 2: Circular Seating and Symmetry

Since the chairs are arranged in a circle, rotating the seating arrangement doesn't create a new arrangement. This means the total number of seating arrangements will be reduced by a factor of 6 to account for rotational symmetry.

Step 3: Placing the men

We are tasked with seating 3 men in a way that no two men sit opposite each other. So, we need to seat the men in such a way that they do not occupy pairs of chairs directly across from one another.

Step 3.1: Choosing positions for the men

We can think of this problem as selecting 3 positions for the men in the 6 available chairs, but with the restriction that no two men can be seated in opposite positions.

If we place one man in any chair, the opposite chair is now forbidden for the remaining two men. So, we effectively need to select 3 chairs such that no two chairs are opposite each other.

• If we select any chair for the first man, there are 3 remaining chairs in which the men can sit. These chairs are not directly opposite any of the other selected chairs. Thus, the men can be seated in the non-opposite chairs.

The number of ways to choose 3 non-opposite chairs out of the 6 available is equivalent to choosing 3 positions from the 6 chairs, excluding the opposite ones. This can be done in 2 ways (either in one of the "even" chair positions or one of the "odd" chair positions).

Step 3.2: Arranging the men

Once we have selected the 3 chairs for the men, the men can be arranged in these 3 chosen seats in 3! ways.

Step 4: Placing the women

After the men are seated, the remaining 3 chairs are occupied by the women. Since there are no restrictions for the women, they can be seated in the remaining chairs in 3! ways.

Step 5: Total number of arrangements

The total number of seating arrangements is then given by: $\text{Total arrangements} = 2 \times 3! \times 3! = 2 \times 6 \times 6 = 72$

Final Answer:

There are 72 ways to seat 3 men and 3 women around the circular table such that no two men are opposite each other.

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If you'd like more details or have further questions, feel free to ask! Here are some related questions to explore further:

1. How does the number of seating arrangements change if there are more men or women?
2. How does the problem change if we remove the restriction on the men sitting opposite each other?
3. What if the table had 8 chairs instead of 6? How would the number of seating arrangements change?
4. How do you calculate the number of ways to arrange people on a circular table without restrictions?
5. What is the formula for seating arrangements on a circular table with restrictions?

Tip: Always account for rotational symmetry when dealing with circular seating problems; it reduces the number of unique seating arrangements.