ISL 1999 C1

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

$$\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.$$

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