Question

On a railway there are 20 stations. The number of different tickets required in order that it may be possible to travel from every station to every station is
A. 210
B. 225
C. 196
D. 105

Answer 1

Hint: We will first find the number of ways in which we can travel from first station to the rest of 19 stations and then using this we will find the tickets required for travelling from first station to rest of the station and that would be 20 including the ticket for the station that we are on and similarly the tickets required for the next station will be 1 less than the previous one because we have already counted a tickets from first station to the second one and similarly we will find the tickets for all the stations then we will use the counting principle to find the number of different tickets required.

Complete step-by-step answer:
Now, we have that on a railway there are 20 stations and we have to find the number of different tickets required in order that it may be possible to travel from every station to every station.
Now, we let the 20 stations be A, b, C, D,……………T.
Now, from station A there are 19 stations to go. So, we have 20 different tickets, so we require a ticket for A also.
Now, similarly we have the different tickets from each station as 17, 16……..1.
Therefore, the total different tickets for 20 stations is $20+19+18+17+16+.......+1$.
Now, we know that the sum of first n natural number that is,
\[1+2+3+..........+n=\dfrac{n\left( n+1 \right)}{2}\].
So, we have the sum of,
\[\begin{align}
  & 1+2+3+..........+20=\dfrac{20\left( 20+1 \right)}{2} \\
 & =210 \\
\end{align}\]
Hence, the correct option is (A).

Note: It is important to note that we have assumed that the phrase from every station to every station means that it is possible to travel for A to B not necessarily from B to A. So, if we consider that then there will be 19 tickets per station. So, in all $20\times 19=380$ that will be the answer.