Question

A train is going from London to Cambridge stops at 12 intermediate stations. 75 persons enter the train during the journey with 75 different tickets of the same class. The number of different tickets they may be holding is
(A) $^{78}{{C}_{3}}$
(B) $^{91}{{C}_{75}}$
(C) $^{84}{{C}_{75}}$
(D) None of these

Answer 1

Calculate the total number of possible ways to reach Cambridge, starting from every 12 stations and also including London as the starting point. It is given that the total number of tickets is 75. Using the combination formula, we know that number of different tickets is $^{m}{{C}_{n}}$, where m is the total number of ways and n is the total number of the number of tickets.

Complete step-by-step solution:
According to the question, it is given that the train is going from London to Cambridge and stops at 12 intermediate stations between them.
We can observe that the can person can start from London can reach Cambridge directly. Another way to reach Cambridge is that he can start in London and stop at the ${{1}^{st}}$ intermediate stop. Then, again start from the ${{1}^{st}}$ intermediate stop and reach Cambridge. In this way, we can say that a person starting from London will face 12 stops at intermediate stations and will have 13 ways to reach Cambridge.
Similarly, the person starting from the next station to London will face 11 stops at intermediate stations and will have 12 ways to reach Cambridge.
Similarly, the person starting from ${{2}^{nd}}$ station to London will face 10 stops at intermediate stations and will have 11 ways to reach Cambridge.
Similarly, the person starting from ${{3}^{rd}}$ station to London will face 9 stops at intermediate stations and will have 10 ways to reach Cambridge.
Similarly, the person starting from ${{4}^{th}}$ station to London will face 8 stops at intermediate stations and will have 9 ways to reach Cambridge.
Similarly, the person starting from ${{5}^{th}}$ station to London will face 7 stops at intermediate stations and will have 8 ways to reach Cambridge.
Similarly, the person starting from ${{6}^{th}}$ station to London will face 6 stops at intermediate stations and will have 7 ways to reach Cambridge.
Similarly, the person starting from ${{7}^{th}}$ station to London will face 5 stops at intermediate stations and will have 6 ways to reach Cambridge.
Similarly, the person starting from ${{8}^{th}}$ station to London will face 4 stops at intermediate stations and will have 5 ways to reach Cambridge.
Similarly, the person starting from ${{9}^{th}}$ station to London will face 3 stops at intermediate stations and will have 4 ways to reach Cambridge.
Similarly, the person starting from ${{10}^{th}}$ station to London will face 2 stops at intermediate stations and will have 3 ways to reach Cambridge.
Similarly, the person starting from ${{11}^{th}}$ station to London will face 1 stops at intermediate stations and will have 2 ways to reach Cambridge.
Similarly, the person starting from ${{12}^{th}}$ station to London will face 0 stops at intermediate stations and will have 1 way to reach Cambridge.
The total number of ways to reach Cambridge = $^{13}{{C}_{2}}=\dfrac{13\times 12}{1\times 2}=78$ .
The total number of tickets = 75.
We know the combination formula that the number of different tickets is $^{m}{{C}_{n}}$ , where m is the total number of ways and n is the total number of tickets.
We have 75 tickets and 78 ways to reach Cambridge.
So, the number of a different set of tickets = $^{78}{{C}_{75}}{{=}^{78}}{{C}_{78-75}}{{=}^{78}}{{C}_{3}}$ .
Hence, the correct option is an option (A).

Note: In this question, one can find the number of ways to reach Cambridge from London and then conclude the answer which is wrong. As the train also stops at 12 intermediate stations, so we have to think about the possible number of ways to reach Cambridge, starting from each intermediate station.