Question

Eleven members of a committee sit round a circular table. In how many ways can they sit so that the secretary and joint secretary are always neighbors of the president?
(a)$\left| \\!{\underline {\, 8 \,}} \right. \times \left| \\!{\underline {\,3 \,}} \right.$
(b)$\left| \\!{\underline {\, {10} \,}} \right.$
(c)$\left| \\!{\underline {\, 8 \,}} \right. \times \left| \\!{\underline {\, 2 \,}} \right.$
(d)$\left| \\!{\underline {\,7 \,}} \right. \times \left| \\!{\underline {\, 2 \,}} \right.$

Answer 1

Here, we will use the concept of circular arrangements to solve this question. We will assume the group of secretary, president and the joint secretary as a single unit to get the total number of people to be arranged. Using the formula of circular arrangements, we will find the required ‘factorial’ keeping in mind that the secretary and the joint secretary can also arrange among themselves.

Formula Used:
When $n$ numbers of people are sitting in a circle, then, the number of ways of arranging them is: $\left( {n - 1} \right)!$

Complete step-by-step answer:
We know that eleven members of a committee sit round a circular table.
Hence, number of members $= 11$
Now, we have to find the number of ways in which they can sit so that the secretary and joint secretary are always neighbors of the president.
Let us assume the secretary, joint secretary and the president to be a single unit.
Number of members other than the three of them $= 11 - 3 = 8$
Now, these 8 members and the single unit which contains the three of them make the total number of people sitting in a circular arrangement as $8 + 1 = 9$
Now, when $n$ numbers of people are sitting in a circle, then, the number of ways of arranging them is: $\left( {n - 1} \right)!$
Hence, the number of ways of arranging 9 people $= \left( {9 - 1} \right)! = 8!$
Since, the secretary and joint secretary are always neighbors of the president, hence, the president would always sit between them.
Therefore, the secretary and the joint secretary can arrange among themselves in $2!$ ways.
Hence, total numbers of ways in which we can arrange those 11 members will be $8! \times 2!$ ways.
Therefore, option C is the correct answer.

Note: In this question, we need to keep in mind that the secretary and the joint secretary can arrange among themselves as well. This small mistake could completely change our answer. Also, while using the formula of circular arrangements, we could forget to count the single unit which consists of the group of president, secretary and the joint secretary. Hence, it must be considered while counting the total number of people to be arranged in a circle. Multiplying both the factorials would give us the required answer.