Question

Between Shirdee and Khandala, there are 10 intermediate stations. The number of ways in which a train can be made to stop at 4 of these stations so that no two of these halting stations are consecutive is:

• A. 35
• B. 70
• C. 105
• D. 10

Answer 1

The correct option is A 35

Let $x_1$be the number of stations before the first halting station, $x_2$ between first and second, $x_3$ between second and third, $x_4$ between third and fourth and $x_5$ on the right of the fourth station, then
$x_1\ge 0,x_2,x_3,x_4\ge 1and{x}_5\ge 0$
such that $x_1+x_2+x_3+x_4+x_5=6$
Now, the total number of ways is the number of solutions of the above equation.
Let $y_2\ge 0,y_3\ge 0,y_4\ge 0$
$\therefore x_1+x_2+x_3+x_4+x_5=6\Rightarrow x_1+(y_2+1)+(y_3+1)+(y_4+1)+x_5=6\Rightarrow x_1+y_2+y_3+y_4+x_5=3$
$\therefore$ The number of solutions ${=}^{3+5-1}{C}_{5-1}{=}^7{C}_4=35$