Question

Between two junction stations $A$ and $B$ , there are $12$ intermediate stations. Find the number of ways in which a train can be made to halt at $4$ of these stations so that no two of these are consecutive

Answer 1

Let $x_{1}:$ the number of stations before first
halting station, after $A$ .
$x_{2}:$ the number of stations between first and second halting stations.
$x_{3}:$ the number of stations between second and third halting stations.
$x_{4}:$ the number of stations between third and fourth halting stations.
$x_{5}:$ the number of stations after the fourth halting station before $B$ .
Then $x_{1}\geq 0,x_{5}\geq 0,x_{2}\geq 1,x_{3}\geq 1,x_{4}\geq 1$
Such that
$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=8$
Let $x_{2}=1+a,x_{3}=1+b,x_{4}=1+c$
Then (i) reduces to
$x_{1}+\left(\right.1+a\left.\right)+\left(\right.1+b\left.\right)+\left(\right.1+c\left.\right)+x_{5}=8$
$\Leftrightarrow x_{1}+a+b+c+x_{5}=5$
$x_{1},x_{5},a,b,c\geq 0$ ....(ii)
Number of non-negative integer solutions of (ii) is same as the number of integer solutions of (i).
$\Rightarrow $ The required number $=^{5 + 5 - 1}C_{5 - 1}$
$=^{9}C_{4}=126$