Question

Rajdhani Express going from Bombay to Delhi stops at five intermediate stations, 10 passengers enter the train during the journey with 10 different tickets of two classes. The number of different sets of tickets they may have is
$
  (a){}^{15}{C_{10}} \\
  (b){}^{20}{C_{10}} \\
  (c){}^{30}{C_{10}} \\
  (d)None\,\,of\,\,these \\
$

Answer 1

Hint: : Find the total possible tickets that are possible for both classes from the total given intermediate trains and use combinations.
We have given 10 passenger boarding the train at the 5 intermediate stations with 10 different tickets of two different classes, so we have to find the different set of tickets of the intermediate stations, so as the passengers start from the first intermediate ticket, they can have 5 possible tickets formed and as the intermediate stations decrease, the possible tickets also decrease, this way we get total of ways for one class and then double for both the classes. With that, we can find the total possible combinations of tickets asked in the question.

Complete step-by-step answer:
For one class, the number of tickets possible from the first intermediate station is 5. From the current and first intermediate station, the maximum possible destinations are five(4 intermediate stations and the final destination), hence, therefore, the maximum number of tickets is 5 and as the train moves forward and reaches the further intermediate station, the possible tickets decrease, which gives us.
$5 + 4 + 3 + 2 + 1 = 15$.
This is for one class, now for both the classes, we have to double the possible tickets for one class, then we will get 30 possible tickets.
So, to find the total number of ways of combinations of sets of tickets are ${}^{30}{C_{10}}$, which is option (c).
So, the correct answer is “Option C”.

Note: We have to find the total possible set of tickets for both the classes and not for one class, so we should not forget to double the ticket possibility from 15 to 30 and we should know that the ticket possibility decreases as we near the destination.