On a problem of Gy. Hajós (Q1096635)

\textit{Gy. Hajós} [Mat. Fiz. Lapok 50, 174 (1943)] proposed the following problem: A sequence \((v_ 1,v_ 2,...,v_{n-1})\) is chosen at random from the \(n^{n-1}\) variations with repetitions of size n-1 of the integers 1,2,...,n. What is the probability that we can find a permutation \((p_ 1,p_ 2,...,p_{n-1})\) of the integers 1,2,...,n-1 such that \(v_ i\leq p_ i\) for all \(i=1,2,...,n-1 ?\) In this note, the author gives a solution and a generalization of this problem. The solution is based on a generalization of the classical ballot theorem.