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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?
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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