Question

There are 24 railway stations between Nellore and Hyderabad. The number of different kind of single second class to be printed so as to enable a passenger to travel from one station to another is
\[\begin{align}
  & A.{}^{25}{{P}_{25}} \\
 & B.{}^{26}{{P}_{2}} \\
 & C.{}^{27}{{P}_{2}} \\
 & D.{}^{28}{{P}_{2}} \\
\end{align}\]

Answer 1

Hint: We are given that, between Nellore and Hyderabad there are 25 stations, so to find the ticket printed, we first find the total station there to choose from. Now, we use ${}^{n}{{\operatorname{P}}_{r}}$ formula to find total number of tickets to be printed, here n is total station while r is the number of station used in one journey, once we have n and r we will get our required result.

Complete step by step answer:
We are given that there are a total of 25 railway stations between Nellore and Hyderabad.
We are asked to find the number of different kinds of single second class to be printed that allows passengers to move from one location to another.
To find such a ticket, we first have to find the total number of stations available for the passenger to choose from.
There are 25 stations between Nellore and Hyderabad. Means 25 stations are there apart from Nellore and Hyderabad. Hyderabad and Nellore are two stations apart from 25 that lie in between them. So, as we are restricted, each ticket must be different from one another. So, repetition of station is not allowed.
We know the formula to find number of choices available if digit is not repeated is given as ${}^{n}{{\operatorname{P}}_{r}}$
Where n is the total number of options and r is the option used at once.
We have, n = 27 and as in one ticket 2 station are used. So, r here is 2. Therefore, r = 2.
So, we get here total number of second class ticket to be printed are ${}^{27}{{P}_{2}}$
Hence correct option is C: ${}^{27}{{P}_{2}}$

Note:
Error like using formula of combination which is given as ${}^{n}{{C}_{r}}$ can happen.
Also remember 25 are the stations between Nellore and Hyderabad, it is not a total station. So, abruptly choosing ${}^{25}{{P}_{2}}$ as an option is an incorrect option. First we add 25+2=27 that give us the total station, then we find the total ticket printed.
2 stations are used at a time because one is an arrival station and the other is a departure station for the train.