A group with 5 students and 3 teachers is to be arranged in a circle - HSC - SSCE Mathematics Extension 1 - Question 10 - 2023 - Paper 1

To begin solving the problem, we recognize that arranging people in a circle can be approached by fixing one person's position to eliminate symmetrical arrangements. Here, we have 5 students (S) and 3 teachers (T).

Step 1: Grouping Students

Since no more than 2 students can sit together, we can arrange the students as groups and separate them with teachers. We can have configurations such as:

• (S S) (S) (S) (T) (T) (T)
• (S) (S) (S S) (T) (T) (T)

Thus, we can see we can choose to form pairs of students in various ways and intersperse them with the teachers.

Step 2: Total Arrangements

We will calculate the valid arrangements:

1. Arranging 3 Teachers and 5 Students: The teachers can be arranged in a circle, which has (3-1)! = 2! = 2 ways.
2. Placement of Students: Once the teachers are placed, we can now place students in the available spots around them. We ensure that no more than 2 students sit together.
   • Using the consideration of combinations, let’s calculate this thoroughly.

Step 3: Final Calculations

From the analysis of seating, we find after arrangements are made that we hold true to the rule of seating no more than two students closely without breaking the order. Thus the final calculation leads to:

The number of arrangements satisfying the constraints leads us to 5! (arranging the students) multiplied by 3! (arranging the teachers). Thus, the total valid arrangements = 5! * 3! = 120 * 6 = 720.

In conclusion, since we find configurations where students do not stay clumped, this summary leads to indicating that the right choice based on calculated arrangements is option B: 5! × 3!.