Question

There are 14 railway stations along a line. Number of ways of selecting 3 stations out of them to stop such that two stops are adjacent is
$\begin{aligned} & a)120 \\\ & b)220 \\\ & c)320 \\\ & d)420 \\\ \end{aligned}$

Answer 1

Now first we will calculate the number of ways in which we can select the two stations such that the two stations are adjacent stops. Now suppose we have selected a pair of adjacent stations then we will check the total number of ways in which we can select the third station such that only two stations are adjacent. Hence we will multiply both the number of possibilities to find the final answer.

Complete step-by-step solution:
Now first let us say we have 14 railway station numbered 1, 2, 3,… 14
Now first we will find the number of ways to select 2 adjacent stations.
Now since the train starts from 1 we cannot select 1 as a stop. Hence the adjacent stops will be
$\left( 2,3 \right),\left( 3,4 \right),\left( 4,5 \right),...\left( 13,14 \right)$
Hence we can count and say that we have 12 adjacent stations where trains can't stop.
Now in the given problem we want the train to stop at 3 stations such that only 2 are adjacent.
Now suppose we have selected the stations $\left( 2,3 \right)$ as adjacent stations. Then the third station can be anything but station number 1, 2, 3 or 4. Hence to select the third station we have 14 – 4 options.
Hence for each adjacent pair selected we can choose the third station in 10 ways.
Hence the total number of ways of selecting 3 stations is $12\times 10$
Hence the number of ways of selecting 3 stations is 120.

Note: Now suppose there are two events A and B and the number of ways of happening event A is m and the number of ways of happening event B is n. then the number of ways of happening event B and event A is $m\times n$ and the number of ways of happening event A or event B is m + n. Here since we want to select 2 stations and then also select a third station then we will multiply the number of possibilities.