We move on a grid by taking only unit steps to the right or up. What is the number of ways to go from the point $(0,0)$ to the point $(m, n)$?

![](https://cdn.mathpix.com/cropped/2024_05_10_699beca3fc1a5ab13db9g-59.jpg?height=263&width=325&top_left_y=1489&top_left_x=781)
The set of paths from $(0,0)$ to $(m, n)$ is in bijection with the set of words of $m+n$ letters written with the letters $\mathrm{D}$ (one step to the right) and $\mathrm{H}$ (one step up) and containing exactly $m$ times the letter $D$ (and therefore $n$ times the letter $H$). For example, the path represented in the figure corresponds to the word DDHDDHHDDHDDHHHDDH. The answer is therefore $\binom{m+n}{m}$.