Give a combinatorial definition of the binomial coefficient $\left(\begin{array}{l}n \\ m\end{array}\right)$ for any non-negative integers $n, m$.

Prove that $\left(\begin{array}{c}n \\ m\end{array}\right)=\left(\begin{array}{c}n \\ n-m\end{array}\right)$ for $0 \leq m \leq n$.

Prove the identities

$$\left(\begin{array}{l} n \\ k \end{array}\right)\left(\begin{array}{l} k \\ l \end{array}\right)=\left(\begin{array}{l} n \\ l \end{array}\right)\left(\begin{array}{l} n-l \\ k-l \end{array}\right)$$

and

$$\sum_{i=0}^{k}\left(\begin{array}{c} m \\ i \end{array}\right)\left(\begin{array}{c} n \\ k-i \end{array}\right)=\left(\begin{array}{c} n+m \\ k \end{array}\right)$$