Question

There are 10 railway stations between station X and another station Y. Find the number of different tickets that must be printed so as to enable a passenger to travel from one station to any other.

Answer 1

Hint: Here 10 stations between x and y means the total number of stations is 12. a passenger will be at any of one station and he has 11 stations left from them he has to choose. That means from 12 stations we have to find a number of ways of arranging 2 stations.

Complete step-by-step answer:
10 railway stations are in between X and Y. So a total of 12 stations are present involving X and Y.
Now, if a person buys a ticket, say from station 1 then he has 11 option to choose since a ticket is not bought for the same station. So, for station 1, there are 11 options. Similarly there are 11 options each for the other 11 stations left.
In all we have 12 stations each having 11 options
Formulating it, we say, there are 12 x 11 possibilities.
Another way to look at this is by using the permutation formula.
We have 12 stations out of which any 2 stations are to be arranged.
Therefore in the formula:
 ${}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}$
n=12 and r=2.
Substituting the values, we get:
$
  {}^{12}{P_2} = \dfrac{{12!}}{{(12 - 2)!}} \\
   = \dfrac{{12!}}{{10!}} \\
   = \dfrac{{12 \times 11 \times 10!}}{{10!}} \\
   = 12 \times 11 \\
   = 132 \\
 $

The answer is 132.

Note: Any of the above methods could be used, depending on which strikes first. In problems like these where possible arrangements are asked, the permutation formula is applied.