Between two junction stations A and B there are 12 intermediate stations. The number of ways a train can be made to stop at 4 of these stations so that no two of these halting station are consecutive is

Options

A. `""^(8)C_(4)`
B. `""^(9)C_(4)`
C. `""^(12)C_(4)-4`
D. none of these

Explanation

Let `x_(1)` be the number of stations before the first halting station, `x_(2)` between first and sencond, `x_(3)` between second and third, `x_(4)` between third and fourth and `x_(5)` on the right of the 4th stations. Then,
`x_(1)ge0,x_(5)ge0,x_(2),x_(3),x_(4)ge1` such that
`x_(1)+x_(2)+x_(3)+x_(4)+x_(5)=8" "....(i)`
The total number of ways is the number of solutions of the above equation.
Let `y_(2)=x_(2)-1,y_(3)=x_(3)-1,y_(4)=x_(4)-1`. Then equation (i) reduces to `x_(1)+y_(2)+y_(3)+y_(4)+x_(5)=5`, where `y_(2),y_(3),y_(4)ge0`. The number of solutions of this equation is `""^(5+5-1)C_(5-1)=""^(9)C_(4)`.