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!(18), 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 is ${}^3{P}_3\times {}^6{P}_6=3!\times 6!$
Total arrangements $=3\times 3!\times 6!$
$=18\times 6!$
$\therefore k=6$