In an examination hall, there are four rows of chairs. Each...

In an examination hall, there are four rows of chairs. Each row has 8 chairs one behind the other. There are two classes sitting for the examination with 16 students in each class. It is desired that in each row all students belong to the same class and that no two adjacent rows are allotted to the same class. In how many ways can these 32 students be seated? (a) $2 \times 16!\times 16$ ! (b) $2 \times 15!\times 15$ ! (c) $2 \times 161 \times 151$ (d) None of these

In an examination hall, there are four rows of chairs. Each row has 8 chairs one behind the other. There are two classes sitting for the examination with 16 students in each class. It is desired that in each row all students belong to the same class and that no two adjacent rows are allotted to the same class. In how many ways can these 32 students be seated?
(a) $2 \times 16!\times 16$ !
(b) $2 \times 15!\times 15$ !
(c) $2 \times 161 \times 151$
(d) None of these

Key Concepts

Counting with Constraints

Counting with constraints involves determining the number of arrangements or selections that satisfy specific conditions or restrictions. This often includes restrictions on positioning or grouping that must be followed, requiring the problem solver to carefully account for these limitations when applying the standard combinatorial tools like permutations and combinations.

Multiplication Principle

The multiplication principle is a core concept in combinatorics which states that if one event can occur in a number of ways and a second independent event can occur in another number of ways, then the total number of outcomes for both events is the product of the individual counts. This principle allows the breakdown of complex count problems into a series of simpler, independent steps whose counts are multiplied together.

Permutations

Permutations refer to the arrangement of items in a specific order. In combinatorial problems, calculating permutations involves determining the number of ways a set of distinct objects can be ordered linearly. This is fundamental in seating problems where each distinct order or arrangement matters, and factorial notation (n!) is often used to represent the total number of arrangements of n items.

Combinations

Combinations deal with selecting items from a larger set where the order of selection does not matter. In many seating or grouping problems, it is necessary to choose subsets (for example, selecting particular individuals for a specific seat grouping) before arranging them, and combinations are mathematically expressed using binomial coefficients such as C(n, k) or 'n choose k'.

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