Three boys and three girls are to be seated around a table...

Three boys and three girls are to be seated around a table in a circle. Among the boys, $X$ does not want any girl neighbour and the girl $Y$ does not want any boy neighbour. How many such arrangements are possible?
(a) 6
(b) 4
(c) 8
(d) 2

Key Concepts

Circular Permutations

This concept involves arranging objects in a circle, where the focus is on the relative order rather than absolute positions. In circular arrangements, rotations are considered equivalent, which means one fixed reference point is typically chosen to simplify counting and eliminate redundant configurations.

Fixing a Reference Point

In problems involving circular permutations, a common technique is to fix one element at a specific position. This anchors the arrangement and removes the circular symmetry, ensuring that arrangements counted are unique with respect to rotations.

Permutations with Restrictions

This refers to counting arrangements that must satisfy specific conditions, such as certain elements not being neighbors or having to be in particular positions. In these problems, additional rules are imposed on the general permutations, often requiring careful case analysis or the application of complementary counting techniques to account for the imposed constraints.

Adjacency Constraints in Seating Arrangements

This key concept deals with problems where certain individuals must or must not be seated next to others. When restrictions are present, standard permutation formulas are modified to ensure that these neighbor conditions are met. Often, the problem must be broken down into cases where the positioning of restricted individuals is considered explicitly, ensuring that prohibited adjacencies are avoided.

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