Question

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

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

Step-by-step solution

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.

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.

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: \( 20! - (19! \cdot 2!) \)

Simplify the expression

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

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) \(18 \cdot 19 !\).

Key concepts

Round Table Conference Seating

In a round table conference, arranging delegates around a circular table presents a unique challenge compared to linear seating arrangements. Unlike a straight line, a round table doesn't have a "first" or "last" seat. This means that when counting permutations for a circular seating arrangement, we must account for the circular nature by fixing one person in place to eliminate symmetrical duplicate arrangements. Thus, the formula used is \((n-1)!\) permutations rather than \(n!\) for linear seating.

• This method considers the rotation of the seating but not reflections, where seating arrangements are the same, just reversed.
• This approach is common in problems involving cyclic symmetry, like arranging keys on a keychain.

Factorial Multiplication

Factorial multiplication is a fundamental concept in permutations and combinations. The factorial of a number, represented as \(n!\), is the product of all positive integers up to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

• In permutations, \(n!\) denotes the total number of ways to arrange \(n\) distinct items.
• It simplifies complex processes of counting arrangements and ensures each possibility is considered.
• In our problem, the total permutations of 20 delegates are calculated using \(20!\). Removing restrictions, we analyze the seating as if all positions are unique and independent.

Combinatorial Restrictions

When solving permutation problems, often times there are specific restrictions, like ensuring certain individuals do not sit together. This adds an additional layer to our calculations.

• First, calculate the unrestricted permutations using the factorial method, giving the total seating possibilities.
• Next, calculate the restricted scenarios separately. Here, for two delegates sitting together, treat them as a single block (so the round seating becomes \((n-1)!\) with them blocked together).
• Finally, use these calculations to determine the permissible arrangements by subtracting the restricted cases from the total.

In our exercise, the calculation adapts by incorporating blocks to account for exclusion of two delegates from sitting together.

Blocking Strategy in Permutations

The blocking strategy is a clever approach to handle permutations with conditions. By treating two or more elements as a single entity, we simplify the challenge of calculating arrangements that meet certain criteria.

• This is done by grouping the restricted elements and considering them as one unit or 'block.'
• For instance, if two delegates must always sit next to each other, they are regarded as one block, thus reducing the number of movable elements.
• The size of your permutation problem effectively becomes one less, simplifying your calculations.

In our task, two delegates are blocked together to compute how it affects their permutation in the round table setup. This helps in efficiently handling exclusions in combinatorial problems and finding valid solutions quickly.