Question

A class has three teachers, Mr. P, 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 $k!\left(18\right),$ then the value of $k$ is

Answer 1

Let T and S denotes 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

In each case the student and the teacher can change their seats among themselves
$\therefore$ for each case total arrangements
$=3!\times 6!$
Total arrangements
$=3\times 3!\times 6!=18\times 6!\therefore k=6$