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If there are 25 railway stations on a railway line, how many types of single second class tickets must be printed so as to enable a passenger to travel from one station to another station?

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: It is given in the above problem that there are 25 railway stations on a railway line and we have to print single second class tickets in such a way so that a passenger can travel from one station to another station. Now, the possible tickets are the total number of combinations of 25 railway stations in such a way that we have to take any two railway stations at a time.

Complete step-by-step solution:
There are 25 railway stations on a railway line and we have to find the number of printed single second class tickets in such a manner that a passenger can travel from one station to another station.
Basically, we have to write the combinations of any 2 railways stations from 25 railways stations.
In the below, we have shown railway stations by station 1 as ${{S}_{1}}$, station 2 as ${{S}_{2}}$, station 3 as ${{S}_{3}}$ and so on till the last station ${{S}_{25}}$.
${{S}_{1}},{{S}_{2}},{{S}_{3}}........{{S}_{25}}$
Now, first of all, we are going to find the number of single tickets from station 1. If we have to print single tickets from station 1 then the possible stations are:
${{S}_{2}},{{S}_{3}},{{S}_{4}}.........{{S}_{25}}$
So, from station 1 to the other stations which we have shown below 24 tickets are possible.
Now, we are going to find the number of single tickets from station 2. The number of stations from station 2 is:
${{S}_{1}},{{S}_{3}},{{S}_{4}}.........{{S}_{25}}$
If you count the above number of stations you will find the number of stations is 24. Now, you might think how the number of stations is 24 because in the stations written above only station 2 is missing and there are a total of 25 stations and if we subtract one station from 25 we get 24.
Hence, the number of stations from station 2 is 24.
Similarly, you will find that from station 3 also the number of stations is 24.
Now, as you can see that from each station from 25 stations 24 single second class tickets are possible so the total number of single second class tickets is equal to the multiplication of 24 single second class tickets to the total number of railways stations which are 25.
Hence, the total number of single second class tickets so that a passenger can travel from one station to another is:
$\begin{align}
  & 24\times 25 \\
 & =600 \\
\end{align}$
Hence, 600 printed single second class tickets are required to enable a passenger to travel from one station to another.

Note: The alternative way of solving the above problem is as follows:
We have given 25 railway stations and we have to find the number of tickets from any one station to another station so basically, we have to select 2 stations from the 25 railway stations but here, the arrangement is also needed because the 25 railway stations are in a particular order so the number of permutations possible is:
${}^{25}{{P}_{2}}$
We know that,
${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$
Using this relation to expand ${}^{25}{{P}_{2}}$ we get,
$\begin{align}
  & {}^{25}{{P}_{2}}=\dfrac{25!}{\left( 25-2 \right)!} \\
 & \Rightarrow {}^{25}{{P}_{2}}=\dfrac{25!}{23!}=\dfrac{25.24.23!}{23!} \\
\end{align}$
In the above equation, 23! will be cancelled out from the numerator and denominator we get,
$\begin{align}
  & {}^{25}{{P}_{2}}=\dfrac{25.24.1}{1} \\
 & \Rightarrow {}^{25}{{P}_{2}}=25.24=600 \\
\end{align}$
Hence, we have calculated 600 single second class tickets that are enable for a passenger to travel from one station to another among 25 railway stations.