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Consider an $m \times n$ integer lattice. A path from $(0, 0)$ to $(m, n)$ can use steps of $(1, 0)$, $(0, 1)$ or diagonal steps $(1, 1)$. Let $D_{m,n}$ be the number of such distinct paths. Prove that $D_{m,n} = \Sigma_k \begin{pmatrix} m \\ k \end{pmatrix} \begin{pmatrix} n+k \\ m \end{pmatrix}$
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