Question

Number of ways in which $7$ people can occupy six seats, $3$ seats on each side in a first class railway compartment if two specified persons are to be always included and occupy adjacent seats on the same side is $(5!)\times k$, then value of $k$ is

Answer 1

Excluding the two specified people, $4$ others can be selected in ${}^5{C}_4$ ways.
The two adjacent seats can be taken in $4$ ways and the two specified people can be arranged between themselves in $2!$ ways, remaining $4$ people can be arranged in $4!$ ways.
$\Rightarrow {}^5{C}_4\cdot 4\cdot (2!)\cdot (4!)=8(5!)$