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Chapter 5: Problem 399

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

Expert verified
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

01

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 \(11!\).
02

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 \(10!\).
03

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).
04

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 \(10! \times 2\).
05

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) \(9(10 !)\).

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