Three boys and three girls are to be seated around a table, in a circle. Among them, the boy X does not want any girl neighbour and the girls Y does not want any boy neighbour. The number of such arrangements possible is A) 4 B) 6 C) 8 D) None of these

This problem is about arranging people around a circular table with some special conditions.

Key concepts:

• In circular permutations, arranging n distinct people around a round table is (n-1)! because rotating the whole circle doesn't create a new arrangement.
• Here, we have 6 people: 3 boys (including X) and 3 girls (including Y).
• The constraints are:
  • Boy X does not want any girl as a neighbour.
  • Girl Y does not want any boy as a neighbour.

Logical steps:

1. Since X cannot sit next to any girl, both neighbours of X must be boys.
2. Since Y cannot sit next to any boy, both neighbours of Y must be girls.
3. There are only 3 boys and 3 girls, so the seating must be arranged so that X is flanked by boys and Y is flanked by girls.

This means X must be seated between the other two boys, and Y must be seated between the other two girls. So the boys must sit together and the girls must sit together in groups of three.

We use circular permutation rules and then count the number of ways to arrange boys and girls internally, considering the fixed positions of X and Y and their neighbours.

This approach helps us find the total number of valid arrangements.

Imagine you have three boys and three girls sitting around a round table. One boy, called X, does not want to sit next to any girl. Also, one girl, called Y, does not want to sit next to any boy. We need to find out in how many different ways they can sit so that both X and Y are happy. To solve this, we think about how to arrange people in a circle and then apply the special rules for X and Y.

Let's solve step-by-step:

1. Total people: 3 boys (X, B2, B3) and 3 girls (Y, G2, G3).

2. Since X does not want any girl neighbour, both neighbours of X must be boys. So X must be seated between B2 and B3.

3. Since Y does not want any boy neighbour, both neighbours of Y must be girls. So Y must be seated between G2 and G3.

4. This means the boys must sit together as a block of 3 (X, B2, B3), and the girls must sit together as a block of 3 (Y, G2, G3).

5. Now, arrange these two blocks around the table. Since the table is round, and we have two blocks, the blocks can be arranged in 2 ways:

   • Boys block followed by girls block
   • Girls block followed by boys block

6. Inside the boys block, X must be between B2 and B3. The boys are sitting in a circle of 3, so the number of ways to arrange them with X between B2 and B3 is:

   • Fix X in one seat (since circular, fix one person to avoid counting rotations).
   • B2 and B3 must be on either side of X, so only 2 ways (B2 on left and B3 on right or vice versa).

7. Inside the girls block, Y must be between G2 and G3. Similarly, fix Y and arrange G2 and G3 on either side:

   • 2 ways.

8. Multiply all possibilities:

   • Number of ways to arrange blocks: 2
   • Number of ways to arrange boys inside block: 2
   • Number of ways to arrange girls inside block: 2

Total arrangements = 2 × 2 × 2 = 8

Final answer: 8 (Option C)