Discrete-math 12

\begin{matrix} \sum_{i = 0}^{k} (\binom{n}{i}) (\binom{n - i}{2 k - 2 i}) 2^{2 k - 2 i} = (\binom{2 n}{2 k}) \end{matrix}

\begin{matrix} n \leq m \\ \sum_{k = 0}^{n} (- 1)^{k} (\binom{n + 1}{k}) (\binom{m + 1 + n - k}{n - k}) = (\binom{m}{n}) \end{matrix}