Use a shortest-paths argument to prove a combinatorics identity

This graphical representation of a typical staircase pattern in a $n times m$ rectangle:

shows 3 boxes, here represented in red, black, blue, the nature of which we are going to explain using natural coordinates conventions.

How are these boxes determined ? By the fact that each staircase pattern reaches the first level at $y=r$ for a certain $x=a$; in the same way, level $y=r+s$ is reached for a certain $x=a+b$, giving intermediate points with coordinates

$$(a,r), (a+b,r+s) $$

(otherwise said, $a$ is determined by $r$, $b$ is determined by $s$...)

Together with endpoints with coordinates $(0,0), (n,m)$, we have all the diagonal "corners" of the 3 boxes attached to the staircase pattern.

Let us now partition the set of staircase patterns into classes $C_{a,b}$ having the same 3 boxes. Each such class being composed of the concatenation of any staircase pattern in the different boxes, we are in a multiplicative context and

$$Card(C_{a,b})=binom{r+a}{a}binom{s+b}{b}binom{t+c}{c}$$

elements. As it is a partition, it remains to sum all these products to get the total number $binom{m+n}{n}$ of staircase patterns.