Consider a large rectangle of size $m$ by $n$, cut into small squares of side 1.

How many ways are there to go from the bottom left corner to the top right corner, moving only on the grid, and only upwards or to the right?
One must take $m$ steps to the right and $n$ steps to the right, for a total of $m+n$ steps, which can be numbered from 1 to $m+n$ following the path. We consider the function that associates with a path the set of the numbers of the steps to the right: this is a subset of $\llbracket 1, m+n \rrbracket$ of cardinality $n$, so the number of paths is $\binom{m+n}{n}$.