Let's place ourselves on a grid $\mathbb{Z} \times \mathbb{Z}$. We start at the origin and only allow moves of one step to the right or one step up $(+(1,0)$ or $+(0,1))$. How many different paths can we take to get to the point $(m, n)$?
In total, exactly $m$ steps to the right (and $n$ steps up) must be taken. To determine a path, it is therefore sufficient to choose among the $n+m$ steps the $m$ that will be to the right. The number of paths is thus $\binom{n+m}{m}\left(=\binom{n+m}{n}\right)$.

## - Tilings and Colorings -

For tiling exercises, there are generally two possible situations: either the tiling requested is possible and one must find a working solution, or it is not possible and one can demonstrate this with a clever coloring. Here are a few examples: