A class consists of three teachers (Mr. P, Ms. Q, Ms. R) and six students (Ali, Bob, Cal, Dee, Amy, Fay). How many ways can they sit in a line of 9 chairs if we have the property that between any two teachers there are exactly two students?
The correct answer is supposed to be 12,960 ways, but I cannot determine this answer. My main assumption is that there are three teachers, the two of these students will have to sit alone on the sides of the teachers. Like so:
S T S S T S S T S
My attempt: $6\times3\times5\times4\times2\times3\times2\times1\times1$ which equals $4320$.