7 women and 7 men are to sit round a circulartable such that there is a man on either side ofevery women. The number of seating arrangements is

Options

A. `(7!)^(2)`
B. `(6!)^(2)`
C. `6!xx7!`
D. 7!

Answer

To solve the problem of seating 7 women and 7 men around a circular table such that there is a man on either side of every woman, we can follow these steps:

### Step 1: Arranging the Women
Since the arrangement is circular, we can fix one woman to eliminate the effect of rotations. The remaining 6 women can then be arranged around the table. The number of ways to arrange \( n \) objects in a circle is given by \( (n-1)! \).

For 7 women, the arrangement in a circle is:
\[ (7-1)! = 6! = 720 \]

### Step 2: Arranging the Men
Once the women are seated, there will be 7 gaps between them (one gap between each pair of women) where the men can sit. Since we have 7 men and 7 gaps, we can arrange the men in these gaps.

The number of ways to arrange 7 men is:
\[ 7! = 5040 \]

### Step 3: Calculating Total Arrangements
To find the total number of seating arrangements, we multiply the number of arrangements of women by the number of arrangements of men:
\[ \text{Total arrangements} = 6! \times 7! = 720 \times 5040 \]

Calculating this gives:
\[ 720 \times 5040 = 3628800 \]

Thus, the total number of seating arrangements is \( 3628800 \).

### Final Answer
The number of seating arrangements is \( 3628800 \).

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