Question

The $10$ students of Batch B feel they have some conceptual doubt on ‘Circular Permutation’. Mr. Sood called them in discussion room and asked them to sit down around a circular table which is surrounded by $13$ chairs. Mr. Sood told that his adjacent seat should not remain empty. The number of ways in which the students can sit around a round table if Mr. Sood also sits on a chair is

• A. ${}^{10}C_2\times {}^{10}C_8$
• B. ${}^{10}C_2\times 2!\times {}^{10}C_8\times 8!$
• C. ${}^{10}C_3\times 3!\times {}^{10}C_7$
• D. None of these

Answer 1

The correct option is B

${}^{10}C_2\times 2!\times {}^{10}C_8\times 8!$

Explanation for the correct option:

Step 1. Find number of ways the students can sit adjacent to the teacher

Given:

Number of chairs $=13$

Number of students $=10$

Number of teacher$=1$

According to the question,

The adjacent two seats to the teacher should not be empty

Then out of $10$ students , the number of ways to choose two students who will sit in these adjacent seats $={}^{10}C_2\times 2!$ ways

Step 2. Find required number of possible ways:

Now, for rest of the $10$ chairs and $8$ students, total number of possible ways $={}^{10}C_8\times 8!$

$\therefore$The number of ways students can sit around the table $\mathbf{=}\mathbf{}{}^{\mathbf{10}}{\mathbf{C}}_{\mathbf{2}\mathbf{}}\mathbf{\times }\mathbf{}\mathbf{2}\mathbf{!}\mathbf{}\mathbf{\times }\mathbf{}{}^{\mathbf{10}}\mathit{C}_{\mathbf{8}}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{8}\mathbf{!}$

Hence, Option ‘B’ is Correct.