Question

Four men and three women are to be arranged at a round table. The number of arrangements in which no two women are together is ....

• A. \(7!\)
• B. \(3!\times 4!\)
• C. \(3!\times 3!\)
• D. \(6!\)

Hint:

Seat the men first around the table, then place the women in the gaps.

Answer 1

The Correct Option is B

Step 1: Fix a Reference:
Fix one man to remove rotation. The other 3 men can be arranged in $3!$ ways.

Step 2: Gaps:
The 4 men leave 4 gaps. Women must occupy 3 different gaps, with order mattering: $4\times3\times2=24=4!$.

Step 3: Total:
$3!\times4! = 144$, option (B).

Final Answer:
Option (B). \[ \boxed{\text{(B) } 3!\times 4!} \]