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: Understanding the Concept:
In a circular arrangement of \(n\) distinct people, positions are relative, giving \((n-1)!\) arrangements.

Step 2: Seat the Men:
Four men can be seated around the round table in \((4-1)! = 3! = 6\) ways.

Step 3: Place the Women:
Four men create exactly 4 gaps between neighbours. To keep women apart, each woman goes into a different gap.
Choose 3 of the 4 gaps and arrange the 3 women there: \(^4P_3 = 4!/1! = 24 = 4!\) ways.

Step 4: Multiply:
\[ 3!\times 4! = 6\times 24 = 144 \]
Option (A), \(7!=5040\), counts all linear arrangements, and (D), \(6!=720\), counts all circular arrangements with no condition. Option (C) \(3!\times3! =36\) undercounts the gaps choice.

Final Answer:
The number of arrangements is \(3!\times4!\), option (B). \[ \boxed{\text{(B) } 3!\times 4!} \]