Question

There are 12 seats in a row of a cinema theatre which is nearer to the entrance. 4 persons enter the theatre and occupy the seats in that row (Before their entry the seats are vacant). If $n_1$is the number of ways in which they occupy the seats, such that no two persons are together and $n_2$ is the number of ways in which each person has exactly one neighbour $n_1:n_2$ is

• A. $7:2$
• B. $3:2$
• C. $5:4$
• D. None of these

Answer 1

Answer: (A)

$7:2$

(1) There are8 empty seats and 9 gaps.
$\therefore n_1={}^9{P}_4=3024$
(2) Each person has exactly one neighbour.
The 4 persons can be paired like $P_1{P}_2:P_3{P}_4$ (say)
Let$x_1$ be the number of empty seats before $P_1{P}_2$ and
$x_2$ be the number of empty seats between $P_1{P}_2$ and $P_3{P}_4$.
$Let{x}_3$ be the number of empty seats after $P_3{P}_4$.
$x_1+x_2+x_3=8$ with $x_1\ge 0,x_2\ge 1,x_3\ge 0$
Put $y_1=x_t;y_2=x_2-1;y_3=x_3$
Number of solutions of $y_1+y_2+y_3=7$ each $y_1\ge 0$
${=}^9{C}_2$
For each pair 2 can be selected in ${}^4{C}_2$ ways, each pair can be arranged in $\lfloor 2$ ways.
$\therefore$ $n_2{=}^9{C}_2^4{C}_2$ $\lfloor 2\lfloor 2=864$
$\therefore$ $\frac{n_1}{n_2}=\frac{3024}{864}=\frac{7}{2}$