Question

A train is going from London to Cambridge stops at 12 intermediate stations. 75 persons enter the train
after London with 75 different tickets of the same class. Number of different sets of tickets they may be
holding is:​

Answer 1

Given v

Number of intermediate stations = 12

Number of travellers = 75

To find :

Number of sets of tickets can be firmed.

Solution :

If one person starts travelling from London,

he has 13 possible stations to which he can end his journey as 12 are intermediate stations and 1 last station.

If person starts at 2nd station then he will have 12 stations.

Similarly if starts from 3rd station, he will have 11 stations and so on.

till, if person starts from 12th station he will have only one last station to end his journey.

So, total number of options are 13 + 12 + 11 +...+ 1

So by formula of adding n natural numbers where n = 13

we get

Total number of options :

$\bf = \frac{n \times (n + 1)}{2} = \frac{13 \times 14}{2} = 13 \times 7 = 91$

= 91

Number of travellers =75

So, Number of choosing different set of tickets, with the combination formula where n = Total number of options available (here 91)

and r = Number of options to be choosen (here 75)

is :

$\binom{n}{r} = \: ^nC_r \: = \frac{n!}{(n-r)!(r)!} \:$

So,

in this case,

Total possible ways :

$\bf \: = \: ^9 \: ^1C_{75}$