What is the ratio of number of arrangements of $9$ people sitting around an isosceles triangular table and the same $9$ people sitting around an equilateral triangular table with $3$ people sitting on each side if the table in both the cases?
My solution approach :-
Let's consider the case of isosceles triangular table.
Number of ways in which people can be arranged on the unequal side = $^9C_3 \times 3!$
Number of ways in which people can be arranged on the equal side = $^6P_6 = 6!$
Total number of ways = $^9C_3 \times 3! \times ^6P_6 = 9!$
But when I come to the case of equilateral triangular table I do not have fixed point to start with. That's where I am getting stuck. How to count the number of ways in case of equilateral triangular table?
One side question what can be the number of ways in which $n$ people can sit around a $x-$ sided polygon? I asked this question because it will help me generalize the cases for any regular polygon.
Thanks in advance !!!