On a chessboard, a piece can only move one square to the right or up. It starts from the bottom left corner and arrives at the top right corner. How many possible paths are there?
On a $k \times l$ chessboard. The piece must move $k-1$ steps to the right and $l-1$ steps up to reach the opposite corner. There are a total of $k+l-2$ moves to be made. A path can be written as a sequence of arrows pointing to the right or up (example: $\rightarrow \rightarrow \rightarrow \uparrow \uparrow$). This means placing $k-1$ right arrows among $k+l-2$ arrows, which is $\binom{k+l-2}{k-1}$. Note that if we had chosen the up arrows, we would have $\binom{k+l-2}{l-1}$, which is the same thing by the first equality from the course.