There are 12 intermediate stations on a railway line between 2 stations. Find the number of ways a train can be made to stop at 4 of these so that no two stopping stations are consecutive.
Initially I found the maximum allowed stop number for the first stop that satisfies the consecutive station condition.
$$A \quad 1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6 \quad 7 \quad 8 \quad 9 \quad 10 \quad 11 \quad 12 \quad B$$
I can have a stop at stations 8, 10, 12. Hence the train can travel a maximum of 6 stations before coming to its first stop, so number of ways $= 6 \cdot 1 \cdot 1 \cdot 1 = 6$.
Then I shift the first stop to station number 5. Now I have 5 options for stop 1 and 2 possible options for any of the next three stops. Hence number of ways $= 5 \cdot 2 \cdot 3 \cdot 1 = 30$.
Again, I shift the first stop to station number 4. Now I have 4 options for stop 1 and 3 possible options for any of the next 3 stops. Hence number of ways $= 4\cdot 3\cdot 3 = 36$.
Continuing the same logic, I arrive at an answer of 156. But the answer I have with me is 126.
Help appreciated. Thanks in advance.