A class has three teachers, Mr. R Ms. Q, and Mrs. R and six students A, B, C, D, E, F. Number of ways in which they can be seated in a line of 9 chairs, if between any two teachers there are exactly two students, is k x 6!, then the value of k is ______.
Let T and S denote teacher and student, respectively.
Then we have following possible patterns according to question:
(i) T S S T S S T S S
(ii) S T S S T S S T S
(iii) S S T S S T S S T
Hence, total number of arrangements are 3 . (3!)6! = 18 x 6!