Question

There are $12$ intermediate stations between two places A and B. Find the number of ways in which a train can be made to stop at $4$ of these intermediate stations so that no two stopping stations are consecutive

Answer 1

Initiall, let's remove the 4 stopping stations.

Then we are left with 8 non-stopping stations $=12-4$ as shown

(non-stopping stations are marked as 1,2,3......8)

Now there are 9 positions (as marked * in the above figure) to place the 4 stopping stations.

Such that no two stopping stations are consecutive

This can be done in ${}^9{C}_4$ ways

Hence, required number of ways ${=}^9{C}_4$

$=\frac{9\times 8\times 7\times 6}{4\times 3\times 2\times 1}$

$=126$