How can I find the number of the shortest paths between two points on a 2D lattice grid?

combinatorics graph-theory integer-lattices

Any shortest path from $(0,0)$ to $(m,n)$ includes $m$ steps in the x axis and $n$ steps in the y axis; what governs the shape of such a path is the order in which we choose to go to the right and up. Since we have $m+n$ total steps to take, we just need to choose which $m$ will be to the right. This is counted by the binomial coefficient $\binom{m+n}{m} = \binom{m+n}{n}$.

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