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Chapter 5: Problem 416

A round table conference is to be held among 20 delegates of 20 countries. The no. of ways they can be seated if two particular delegates are never sit together is. (a) \(17 \cdot 18 !\) (b) \(18 \cdot 19 !\) (c) \((20 ! / 2)\) (d) \(19 ! \cdot 2\)

Short Answer

Expert verified
The correct answer is (b) \(18 \cdot 19 !\).

Step by step solution

01

Calculate the total number of ways to seat the delegates

First, let's find the total number of ways to seat the 20 delegates (without any restrictions). Since there are 20 seats, there are 20! (20 factorial) possible ways to arrange the delegates.
02

Calculate the number of ways in which the two particular delegates sit together

Now, we want to find the number of ways in which the two particular delegates sit together. We can treat them as a single unit (a "block") and then place the remaining 18 delegates. This will effectively make it 19 positions for the remaining 18 delegates and the block. So, there are 19! ways to arrange the delegates along with the block. However, within the block itself, there are 2! (or 2) ways to arrange the two particular delegates. Therefore, there are \(19! \cdot 2! \) ways where the two delegates sit together.
03

Calculate the number of ways in which the two particular delegates do NOT sit together

To find the number of ways in which the two delegates do NOT sit together, we will subtract the number of ways in which they sit together (from Step 2) from the total number of ways (from Step 1). So, the number of ways in which the two particular delegates do NOT sit together is: \( 20! - (19! \cdot 2!) \)
04

Simplify the expression

Now let's simplify this expression: \( 20! - (19! \cdot 2!) = 19! \cdot (20 - 2) = 19! \cdot 18\)
05

Identify the correct answer

The number of ways in which the two particular delegates do NOT sit together is \(19! \cdot 18\). This matches option (b) in the given choices. Therefore, the correct answer is (b) \(18 \cdot 19 !\).

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