These questions are based on the following information. Five...

These questions are based on the following information. Five boys $-$ Anil, Charan, David, John and Kamal sit in a row facing north, not necessarily in the same order. I. John sits exactly in between Anil and David. II. John sits exactly in between Charan and Kamal. In how many different ways these five boys can sit? (A) 2 (B) 4 (C) 8 (D) 16

These questions are based on the following information.

Five boys $-$ Anil, Charan, David, John and Kamal sit in a row facing north, not necessarily in the same order.
I. John sits exactly in between Anil and David.
II. John sits exactly in between Charan and Kamal.
In how many different ways these five boys can sit?
(A) 2
(B) 4
(C) 8
(D) 16

Key Concepts

Permutation

Permutation refers to the different ways in which a set of items can be arranged in order. In combinatorial problems, the order of arrangement is key, and the total number of arrangements for a set is often given by factorial calculations. When restrictions are imposed, such as positional or relational constraints, the overall count of initial arrangements is reduced by eliminating those that do not meet the criteria.

Constraint-based Arrangement

A constraint-based arrangement involves placing items while adhering to certain rules or restrictions. This is common in problems where specific conditions must be met, such as an element needing to be positioned between two particular others. Such restrictions require tailored adjustments to the typical permutation count, as only a subset of all possible arrangements satisfies the condition.

Relative Positioning

Relative positioning in arrangements deals with the ordering of elements based on their positions in relation to each other. For example, specifying that one element must be exactly between two others affects the valid placements of all those elements. Understanding relative positioning is crucial in solving combinatorial problems where the relationships between items dictate the allowable permutations.

Case Analysis

Case analysis is a systematic approach to solve problems by dividing the situation into distinct scenarios based on the given constraints. Each case is analyzed separately to determine its contribution to the overall count, ensuring that all possible valid arrangements are considered. This method is especially useful when multiple, overlapping conditions need to be simultaneously satisfied.

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