On a railway route between two places A and B there are 10 stations on the way. If 4 new stations are to be added, how may types of new tickets will be required if each ticket is issued for a oneway journey?

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Q: 67 (IAS/2005)
On a railway route between two places A and B there are 10 stations on the way. If 4 new stations are to be added, how may types of new tickets will be required if each ticket is issued for a oneway journey?

question_subject: 

Logic/Reasoning

question_exam: 

IAS

stats: 

0,1,16,5,6,5,1

keywords: 

{'railway route': [0, 0, 1, 0], 'oneway journey': [0, 0, 1, 0], 'new stations': [0, 0, 1, 0], 'new tickets': [0, 0, 1, 0], 'stations': [0, 0, 1, 3], 'ticket': [0, 0, 2, 0], 'types': [11, 3, 20, 23]}

In the initial situation, there are 10 stations. When 4 new stations are added, the total number of stations becomes 14.

Option 1 says only 14 new tickets are required, but this incorrectly considers just the new stations.

Option 2 suggests that 48 new tickets are required, however, this is also incorrect.

Option 3 proposes 96 new tickets. While this doubles the count in option 2, it is still not assuming one-way tickets between each pair of stations.

The fourth option is 108, which is correct. Essentially, this number is calculated by considering that each of these stations can produce one way tickets to all the other stations. In other words, for each of the 14 stations, we have 13 unique one-way tickets (14-1 since a station cannot issue a ticket to itself). This results in a total of 14*13 = 182. However, this counts each ticket twice (A to B, and B to A are the same), so we must halve this number, which gives us 182/2 = 91. This is the base count for the already existing stations. We then have four new stations, each with 14 one-way journeys, giving an additional

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