IMO Shortlist 1999 problem C1

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Dodao/la: arhiva
2. travnja 2012.
Let n \geq 1 be an integer. A path from (0,0) to (n,n) in the xy plane is a chain of consecutive unit moves either to the right (move denoted by E) or upwards (move denoted by N), all the moves being made inside the half-plane x \geq y. A step in a path is the occurence of two consecutive moves of the form EN. Show that the number of paths from (0,0) to (n,n) that contain exactly s steps (n \geq s \geq 1) is

\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.
Izvor: Međunarodna matematička olimpijada, shortlist 1999