How many paths are there on $\mathbb{Z}^{2}$ starting from $(0,0)$, taking steps $+(1,0)$ or $+(0,1)$, and ending at $(m, n)$, where $m, n \geqslant 0$?
Such a path must consist of $\mathfrak{m}+\mathfrak{n}$ steps, including $\mathfrak{m}$ times $+(1,0)$ and $\mathfrak{n}$ times $+(0,1)$. Therefore, it is sufficient to choose the positions of the $m$ steps that are $+(1,0)$ from the $m+n$ possible steps. The total number is thus $\binom{m+n}{m}$.