Arranging Girls and Boys at a Round Table: A Comprehensive Guide

In this article, we will explore the intriguing problem of arranging 3 girls and 3 boys at a round table in such a way that no girl sits directly opposite another girl. This classic problem in combinatorial mathematics highlights the complexity and elegance of circular permutations.

Introduction to the Problem

The task at hand involves solving a circular permutation problem with a specific constraint. The challenge lies in arranging 6 individuals (3 girls and 3 boys) around a table in a manner that satisfies the given condition. This scenario is not uncommon in scenarios such as formal dining arrangements, classroom seatings, or game playing.

Step-by-Step Solution

To solve this problem, we will follow a systematic approach:

Fixing a Position

Since the table is round, we start by fixing the position of one individual to eliminate equivalent rotations. Let's fix one boy (Boys) in position 1.

Arranging the Remaining Boys

After fixing one boy, we have 2 remaining boys to arrange. The number of ways to arrange 2 boys is given by 2! (2 factorial).

2!  2

Determine the Positions for Girls

With one boy fixed and the other two boys arranged, the arrangement looks like this:

- B1 (fixed)- B2- B3

The girls can only sit in the positions that are not opposite each other. In a round table with 6 positions, the opposite positions would be:

Opposite B1: P4Opposite B2: P5Opposite B3: P2

The valid positions for the girls are:

- P2 next to B1- P3 next to B2- P6 next to B3

Arrange the Girls

The 3 girls can occupy these 3 available positions in 3! (3 factorial) ways.

3!  6

Combine the Arrangements

The total number of arrangements is the product of the arrangements of the boys and the arrangements of the girls.

Total arrangements  ways to arrange boys × ways to arrange girls  2! × 3!  2 × 6  12

Thus, the total number of ways to arrange 3 girls and 3 boys at a round table such that no girl sits directly opposite another girl is 12.

Alternative Approach

Another way to approach the problem is to pick a start point and number the seats from 1 to 8 around the circle. The key constraint here is that the number of seats between two consecutive girls cannot be two for all three pairs and cannot be one for all three pairs.

Assigning Seats for Girls

Without loss of generality, we can assign seats 1, 3, and either 5 or 6 for the girls. Now we can permute the boys in the remaining 5 seats and the girls in the assigned seats:

2 × 5! × 3!  2 × 120 × 6  1440

Reference Point Method

Let's use Girl A as the reference point. Going around the table clockwise from her left, we have 5 boys. There are only 4 positions between these boys where we can insert a girl without her being next to Girl A:

So there are 4 ways to place Girl B and 3 ways to place Girl C in the remaining insertion points. Therefore, that's 4 × 3 12 ways to position the girls.

Now we select the boys to arrange them in their positions. There are 5! 120 ways to do this.

Therefore, the total number of arrangements is 12 × 5! 12 × 120 1440.

Conclusion

In this comprehensive guide, we have explored different methods to solve the problem of arranging girls and boys at a round table with specific constraints. Both approaches demonstrate the importance of thoughtfully fixing a position and carefully counting valid arrangements.

The final number of valid arrangements is 1440, as derived from both the systematic approach and the alternative method.

Keywords: round table arrangements, seating arrangements, circular permutations