In how many ways 7 men and 7 women can be seated around a round table such that no two women can sit together

Answer

To solve the problem of seating 7 men and 7 women around a round table such that no two women sit together, we can follow these steps:

### Step 1: Arrange the Men
Since we are arranging the men around a circular table, we can fix one man to eliminate the effect of rotations. The number of ways to arrange the remaining 6 men is given by (n-1)!, where n is the number of men.

\[ \text{Ways to arrange 7 men} = (7 - 1)! = 6! = 720 \]

### Step 2: Identify the Seating Positions for Women
Once the men are seated, they create 7 gaps between them where the women can sit. Since we want to ensure that no two women sit together, each woman must occupy one of these gaps.

### Step 3: Arrange the Women
The number of ways to arrange the 7 women in the 7 gaps created by the seated men is given by the factorial of the number of women.

\[ \text{Ways to arrange 7 women} = 7! = 5040 \]

### Step 4: Calculate the Total Arrangements
To find the total number of arrangements of men and women around the table, we multiply the number of arrangements of men by the number of arrangements of women.

\[ \text{Total arrangements} = 6! \times 7! = 720 \times 5040 = 3,628,800 \]

### Final Answer
Thus, the total number of ways to seat 7 men and 7 women around a round table such that no two women sit together is:

\[ \text{Total arrangements} = 3,628,800 \]

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