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

Answer: 126

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 $x_1\ge 0,x_5\ge 0,x_2\ge 1,x_3\ge 1,x_4\ge 1$
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
$x_1+(1+a)+(1+b)+(1+c)+x_5=8$
$\iff x_1+a+b+c+x_5=5$
$x_1,x_5,a,b,c\ge 0$ ....(ii)
Number of non-negative integer solutions of (ii) is same as the number of integer solutions of (i).
$\Rightarrow$ The required number ${=}^{5+5-1}{C}_{5-1}$
${=}^9{C}_4=126$