Let
![n \geq 1](/media/m/a/9/8/a982fcac3e2c9e0d94e965d6efb5a582.png)
be an integer. A path from
![(0,0)](/media/m/3/e/4/3e48f4571311d9b0074aa083e1ebe311.png)
to
![(n,n)](/media/m/6/2/3/6233cbd5bea84837cb8cad96dc65ff05.png)
in the
![xy](/media/m/5/9/6/596af52a0894be7f886bc10ba63d140f.png)
plane is a chain of consecutive unit moves either to the right (move denoted by
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
) or upwards (move denoted by
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
), all the moves being made inside the half-plane
![x \geq y](/media/m/4/c/f/4cf15742197dfe4cb55aac6dfcd77ba2.png)
. A step in a path is the occurence of two consecutive moves of the form
![EN](/media/m/8/a/7/8a738873fd9bb9b92a40601aa803337f.png)
. Show that the number of paths from
![(0,0)](/media/m/3/e/4/3e48f4571311d9b0074aa083e1ebe311.png)
to
![(n,n)](/media/m/6/2/3/6233cbd5bea84837cb8cad96dc65ff05.png)
that contain exactly
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
steps
![(n \geq s \geq 1)](/media/m/f/0/a/f0a45ce55f9310382e67fe5569133649.png)
is
%V0
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
$$\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.$$