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50 people have voted in an election, in which they are two candidates, and 25 people have voted for one candidate, and 25 people have voted for the other. You don’t know this yet, and are counting the votes, by looking at the ballots in sequence. What is the number of sequences for which, as you count the votes, neither candidate is ever ahead of the other by more than 15 votes, but the end result is a tie?

There is an answer already here, very neat answer: Vote Counting Problem

However, I would like to have a more systematic way to tackle these kind of questions and would like to use generating functions. Would anyone please start me or give me an idea on how to use GF on this problem?

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  • $\begingroup$ The good news is that it is impossible for both candidates to have been ahead by more than $15$ votes in the same count $\endgroup$ Commented Mar 21, 2019 at 0:14
  • $\begingroup$ Do you have any reason to believe there is a GF solution? Have you seen similar problems with a GF solution, for example? $\endgroup$ Commented Mar 21, 2019 at 4:40
  • $\begingroup$ @Henry, if you get all 25 votes for A first, it certainly will have a 15 (or more) advantage for some time... $\endgroup$ Commented Jan 29, 2020 at 20:31
  • $\begingroup$ @vonbrand: If that happens then B will never be $15$ votes ahead. My point was that you did not have to worry about double counting or inclusion/exclusion etc. in this particular example $\endgroup$ Commented Jan 29, 2020 at 23:06

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