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There are 2n people standing in a queue in front of a ticket booking office. n of those 2n people have a Rs 10 note each. Rest of the people have a Rs 5 note each. At first the man at the ticket booking office do not have any money. The price of each ticket being Rs 5, in how how many ways can the queue be arranged so that there is no problem in change?

There are 2n people standing in a queue in front of a ticket booking office. n of those 2n people have a Rs 10 note each. Rest of the people have a Rs 5 note each. At first the man at the ticket booking office do not have any money. The price of each ticket being Rs 5, in how how many ways can the queue be arranged so that there is no problem in change?

Grade:11

1 Answers

mycroft holmes
272 Points
11 years ago

Label the people who have Rs. 5 with them as A and those with Rs.10 as B. You need a sequence of length 2n composed of n A's and n B's such that number of B's never exceeds the number of A's at any segment of the sequence taken from the initial point. That is called in combinatorics as a Dyck Sequence. 

 

The number of such sequences is the nth Catalan Number given by 

 

This is out of your syllabus, but of your interested you can consult the wiki article http://en.wikipedia.org/wiki/Catalan_number

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