A class has tree teachers, Mr. `X`, Ms.`Y` and Mrs.`Z` and six students `A`, B`, `C`, `D`, `E`, `F Number of ways in which they can be seated in a line of 9` chairs,if between any two teachers there are exactly two students is

Options

A. `18xx6!`
B. `12xx6!`
C. `24xx6!`
D. `6xx6!`

Explanation

To solve the problem of seating 3 teachers (Mr. X, Ms. Y, and Mrs. Z) and 6 students (A, B, C, D, E, F) in a line of 9 chairs with the condition that there are exactly 2 students between any two teachers, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
We need to arrange the teachers and students such that there are exactly 2 students between any two teachers. The arrangement can be visualized as:
- Teacher, Student, Student, Teacher, Student, Student, Teacher, Student, Student
This means that the arrangement must start and end with a teacher.

2. **Identifying the Positions**:
In the arrangement, the positions of the teachers and students can be identified as follows:
- T S S T S S T S S
Here, T represents a teacher and S represents a student.

3. **Counting the Ways to Arrange Teachers**:
There are 3 teachers, and they can be arranged among themselves in:
\[ 3! = 6 \text{ ways} \]

4. **Counting the Ways to Arrange Students**:
There are 6 students, and they can be arranged among themselves in:
\[ 6! = 720 \text{ ways} \]

5. **Calculating Total Arrangements**:
Since the arrangement of teachers and students is independent, we multiply the number of arrangements of teachers by the number of arrangements of students:
\[ \text{Total arrangements} = 3! \times 6! = 6 \times 720 = 4320 \]

6. **Final Arrangement**:
Since there are 3 different valid arrangements of the sequence (as shown in the video), we multiply the total arrangements by 3:
\[ \text{Final Total} = 3 \times (3! \times 6!) = 3 \times 4320 = 12960 \]

### Conclusion:
The total number of ways in which the teachers and students can be seated in a line of 9 chairs, with exactly two students between any two teachers, is:
\[ \boxed{12960} \]