Задача 12.

Докажите, что \begin{equation*} \begin{array}{ll} 1) \binom{n}{0}^2+\binom{n}{1}^2+...+\binom{n}{n}^2=\binom{2n}{n};\\ 2) \binom{n}{0}+\binom{n+1}{1}+...+\binom{n+k-1}{k-1}+\binom{n+k}{k} = \binom{n+k+1}{k}; \\ 3) \binom{n}{1}+2\binom{n}{2}+...+n\binom{n}{n}= n2^{n-1};\\ 4) \binom{n}{k}\cdot\binom{n-k}{m-k}=\binom{m}{k}\cdot\binom{n}{m}. \end{array} \end{equation*}