There are p intermediate stations on a railway line from one terminus to another . In how many ways a train can stop at 3 of these intermediate stations if no two of those stopping stations are to be consecutive ?

Explanation

Let there be p intermediate stations betwee two terminus stations A and B as shown above.
Number of ways the train ca stop in three intermediate stations=`.^(p)C_(3)`
these are comprised of two exclusive cases viz.
(i) atleast two stations are consecutive.
(ii) now of which is consecutive.
Now, there are (p-1) pairs of consecutive intermediate stations.
In order to get a station trio in which attleast two stations are consecutive, each pair can be associated with a third station in (p-2) ways. hence, total number of ways in which 3 stations consisting of atleast two consecutive stations, can be chosen in (p-1)(p-2) ways. among these, each triplet of consecutive stations occur twice.
For example, the pair `(S_(n),S_(n-1))` when combined with `S_(n+1)` and the pair `(S_(n),S_(n+1))` when combined with `S_(n-1)` gives the same triplet and is counted twice. so, the number of three consecutive stations trio should be subtracted.
Now, number of these three consecutive stations trio is
(p-2).
Hence, the number of ways the triplet of stations consisting of atleast two consecutive stations can be chosen in
`={(p-1)(p-2)-(p-2)}` ways
`=(p-2)^(2)` ways
Therefore, number of ways the train can stop in three consecutive stations.
`=.^(p)C_(3)-(p-2)^(2)=(p(p-1)(p-1))/(1*2*3)-(p-2)^(2)` ltBrgt `=(p-2)[(p^(2)-p-6p+12)/(6)]=((-2)(p^(2)-7p+12))/(6)`
`=((p-2)(p-3)(p-4))/(1*2*3)=.^((-2))C_(3)`