Question

12 Persons are to be arranged to a round table, If two particular persons among them are not to be side by side, the total number of arrangements is : (a) \(9(10 !)\) (b) \(2(10) !\) (c) \(45(\overline{8 !)}\) (d) \(10 !\)

Short answer

The total number of arrangements where the two specific people are not seated together is \(9(10!)\). The correct answer is (a) \(9(10 !)\).

Step-by-step solution

Calculate total number of arrangements without constraint

Without any constraints, we have a circular arrangement problem. The number of arrangements for n people around a circle is \((n-1)!\). In this case, we have 12 people, so the total number of arrangements will be \((12-1)!\), which is \(11!\).

Calculate the number of arrangements with the two people seated together

To calculate the number of arrangements where the two specific people are seated together, we can treat them as a single unit. This means we will be considering 11 entities around the circular table (10 individuals plus the combined pair). Using the same circular arrangement formula as before, the number of arrangements will be \((11-1)!\), which is \(10!\).

Calculate the number of arrangements for positions within the pair

Now we need to find the number of ways to arrange the two specific people within their pair. Since there are only two people, there can be only 2 ways of arranging them (either Person A is to the left of Person B, or Person B is to the left of Person A).

Calculate the total number of arrangements with the pair seated together

To determine the total number of arrangements with the pair seated together, we need to multiply the number of arrangements from Step 2 and Step 3. This gives us a total of \(10! \times 2\).

Subtract the unwanted arrangements from the total arrangements

Finally, we subtract the number of arrangements with the pair seated together (found in Step 4) from the total number of arrangements without any constraints (found in Step 1). This will give us the number of arrangements where the pair is not seated together: \(11! - (10! \times 2)\) Calculating this expression yields: \(11! - (10! \times 2) = 9(10!)\) Thus, the total number of arrangements where the two specific people are not seated together is \(9(10!)\). The correct answer is (a) \(9(10 !)\).

Key concepts

Factorial Notation

Factorial notation is a fundamental concept in mathematics used to denote the product of all positive integers up to a given number. It is represented by an exclamation mark. For instance, the factorial of a number \( n \) is written as \( n! \). This can be calculated as:

• \( n! = n \times (n-1) \times (n-2) \times \, ... \, \times 1 \)

Factorial notation simplifies the process of counting permutations, especially in combinatorial problems where order is important. In the context of circular permutations, such as arranging people around a table, the formula slightly modifies.

• The number of arrangements of \( n \) people in a circle is given by \( (n-1)! \).

This is because in a circular arrangement, there is no distinct starting point, which reduces the number of possible arrangements by a factor of \( n \). Understanding factorials is crucial for solving more complex problems involving arrangements and constraints.

Arrangement Constraints

In combinatorial problems, we often face constraints that affect how we can arrange elements. Arrangement constraints limit the possible combinations by introducing rules.For example, in our exercise, two specific persons must not sit next to each other. When faced with such constraints, one common method is to first consider the scenario where the constraint does not exist.

• Calculate the total number of configurations without any constraints (e.g., \( (12-1)! \) for 12 people).
• Next, calculate how many configurations violate the constraints (e.g., treating the two constrained individuals as a single unit or block and finding arrangements for this block).
• Finally, subtract the violating arrangements from the total to find the permissible configurations.

This systematic approach ensures that all possibilities are considered, and the solution respects all imposed constraints.

Combinatorial Problems

Combinatorial problems involve counting the number of ways to arrange or select items within a set, often under specific conditions or restrictions. These problems can range from simple to highly complex, depending on the constraints involved and the nature of the elements being arranged.In circular arrangements, one must often deal with limited rotational symmetry. The problem described is a classic example where two individuals should not be adjacent.To solve it, we break the problem into manageable pieces:

• Find the solutions without constraints using factorial notation (\(11!\) for 12 individuals).
• Determine how to account for the constraint (calculate the configurations where the two must sit together).
• Subtract unwanted arrangements from the total possible arrangements to find the solution under constraints (i.e., \(11! - (2 \times 10!)\)).

By applying these techniques, we manage to simplify and effectively solve what initially seems a complicated problem, turning it into a structured and logical process.