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 insert the students, we will consider the arrangement of the 3 teachers creating 3 gaps for students (between each pair of teachers) and one gap at the end. This gives us a total of 4 gaps.

We need to choose gaps to place the students while ensuring no more than 2 can sit together. The possible combinations of inserting students in the gaps include:

• 2 students in one gap and 1 student in each of the remaining gaps.
• This leads to choosing 2 gaps out of the 4 to hold the students, calculated as:

$C(4, 2) = 6$ combinations.

After choosing which gaps will contain the students, we can arrange the 5 students in those selected positions. This is done by permuting the students:

$5! = 120$ ways.

Thus, the total arrangements will be:

$2 (teachers) \times 6 (gaps) \times 120 (students) = 1440$.

Altogether, considering all combinations and arrangements, the final answer is:

The answer would match option B: 5! \times 3!.