Question

10 person are to be arranged around a round table. 3 persons wish to sit as a group number of ways the arrangement can be made is (a) \(9 ! \times 3 !\) (b) \(8 ! \times 3 !\) (c) \(7 ! \times{ }^{8} \mathrm{P}_{3}\) (d) \(7 ! \times 3 !\)

Short answer

The arrangement of 10 people around a round table with 3 specific persons wishing to sit together as a group can be made in \(7 ! \times 3 !\) ways.

Step-by-step solution

Arrange 7 normal persons and 1 group of people around the table

Let's consider the group of 3 persons as a single entity. This makes it a total of 8 entities (7 normal persons + 1 group) to be arranged around the table. Since the round table arrangement will be considered the same if it is rotated together, we will have a total of (8-1)! = 7! ways to arrange these entities.

Arrange 3 persons within their group

Now, within the group of 3 persons, they can be arranged among themselves in 3! ways since there are 3 seats available within the group.

Calculate the total number of ways for the arrangement

To calculate the total number of ways for the arrangement, we need to multiply the number of ways from Step 1 and Step 2. So the total number of ways = (7!) × (3!). The arrangement can be made in 7! × 3! ways. The correct answer is: (d) \(7 ! \times 3 !\)

Key concepts

Circular Permutation

When arranging people or items in a circle, the concept of circular permutation comes into play. Unlike linear arrangements, where the start and end are distinct, a circle has no beginning or end. For example, if you have 8 people, arranging them around a table has the same view regardless if rotated. Therefore, with circular permutations, the number of ways to arrange \( n \) items is \((n-1)!\). This is because one of the positions is fixed to account for the symmetry.

• In our exercise, we treat 7 individuals and 1 group as 8 elements.
• Therefore, the circular permutation of these 8 elements is \((8-1)!=7!\).

Arrangement Problems

Arrangement problems involve finding how many different ways you can arrange a set of items. There can be additional constraints, such as specific items needing to stay together. In circular arrangements, a subgroup can be treated as a single unit or block. This is a common approach to simplify the problem.

• For example, in the given exercise, 3 specific people want to sit as a group.
• This group is treated as a single entity at first, reducing the complexity of the circular arrangement.

This technique helps break down larger arrangement problems into smaller, manageable parts. Within the block, you can determine the arrangements of the internal elements, in this case using a linear permutation for the 3 people inside the group.

Factorial Concepts

Factorials are a fundamental part of permutations and combinations. The notation \(n!\) (read "n factorial") represents the product of all positive integers up to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

In permutation problems:

• Factorials are used to calculate the total number of possible arrangements of \(n\) distinct items.
• When considering the internal arrangements of a grouped block – like the 3 people here – use \(3!\) to determine how many ways they can be arranged among themselves.

In this problem:

• We used \(7!\) for the circular permutation of the entire table.
• We multiplied by \(3!\) to account for the arrangements within the grouped block.

Factorials simplify the calculation of permutations and guarantee accuracy in complex arrangement scenarios.