Suppose
and
are candidates for office and there are voters, voting for and for . In how many ways can the ballots be counted so that is never ahead of ? The solution is a Catalan number .

A related problem also called "the" ballot problem is to let receive votes and
votes with .
This version of the ballot problem then asks for the probability that stays ahead of as the votes are counted (Vardi 1991). The solution is , as first shown by M. Bertrand
(Hilton and Pedersen 1991). Another elegant solution was provided by André
(1887) using the so-called André's
reflection method.

The problem can also be generalized (Hilton and Pedersen 1991). Furthermore, the TAK function is connected with the ballot problem
(Vardi 1991).

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Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht,
Netherlands: Reidel, p. 22, 1974.Feller, W. An
Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed.
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Ballot Problem and Catalan Numbers." Nieuw Arch. Wisk.8, 209-216,
1990.Hilton, P. and Pedersen, J. "Catalan Numbers, Their Generalization,
and Their Uses." Math. Intel.13, 64-75, 1991.Kraitchik,
M. "The Ballot-Box Problem." §6.13 in Mathematical
Recreations. New York: W. W. Norton, p. 132, 1942.Motzkin,
T. "Relations Between Hypersurface Cross Ratios, and a Combinatorial Formula
for Partitions of a Polygon, for Permanent Preponderance, and for Non-Associative
Products." Bull. Amer. Math. Soc.54, 352-360, 1948.Vardi,
I. Computational
Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 185-187,
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