Mathematics - Permutation & Combination Question with Solution | TestHub

The total number of consecutive pairs of stations, namely $S_1{S}_2$, $S_2{S}_3$, ..., $S_{n-1}{S}_n$, is $(n-1)$. Each of these pairs can be associated with a third station in $(n-2)$ ways. Thus, choosing a pair of stations and any third station can be done in $(n-1)(n-2)$ ways.

The above count also includes cases of three consecutive stations. However, each such case has been counted twice. For example, the pair $S_4{S}_5$ combined with $S_6$ and the pair $S_5{S}_6$ combined with $S_4$ are identical. Hence, subtracting the excess counting, the number of ways in which three stations can be chosen so that at least two of them are consecutive is: