A teacher and 12 students (6 boys and 6 girls) are sitting around a circular table. In how many ways can these 13 indiciduals sit if out of the 12 students no two boys or no two girls sit next to each other ?
My attempt :-
Firstly, I made all the 6 boys sit on the 6 chairs such that no 2 of them are together, these 6 boys can be arranged in 6! ways. There will be 6 gaps created in total, in which there will be 1 gap having 2 seats and rest of the 5 gaps will have 1 seat. Now I can place the 6 girls into these 6 gaps in 6! ways, and the last seat will automatically be assigned to a teacher.
so as per my understanding it should be 6!*6! ways
however the official answer is given as 2*6!*6! ways
How is this factor of 2 coming in ?
Best Answer
First fix the teacher as reference at the north seat
Now seating them clockwise, it has either to be
BGBGBGBGBGBG or GBGBGBGBGBGB
Hence the factor of $2$, yielding $2*6!6!$ arrangements