Question

There are five boys and three girls who are sitting together to discuss a management problem at a round table. In how many ways can they sit around the table so that no two girls are together?

• A. 1220
• B. 1400
• C. 1420
• D. 1440

Hint:

Seat the boys first in a circle, then place the girls one per gap between boys so no two girls are adjacent.

Answer 1

The Correct Option is D

Step 1: Arrange the boys around the round table.
For a circular arrangement of \(n\) people, the number of arrangements is \((n - 1)!\). With 5 boys, this gives: \[ (5 - 1)! = 4! = 24 \text{ ways} \]

Step 2: Identify the gaps for the girls.
Once the 5 boys are seated in a circle, there are exactly 5 gaps between consecutive boys. To ensure no two girls sit together, each girl must occupy a separate gap.

Step 3: Choose and arrange the girls in the gaps.
Choose 3 of the 5 gaps for the girls: \(\binom{5}{3} = 10\) ways. Then arrange the 3 girls in these chosen gaps: \(3! = 6\) ways. So the girls can be placed in \(10 \times 6 = 60\) ways.

Final Answer:
Total arrangements = (ways to seat boys) \(\times\) (ways to seat girls) = \(24 \times 60 = 1440\). \[ \boxed{1440} \]