\[\begin{cases} \sum_k \binom{r}{m+k}\binom{s}{n-k}=\binom{r+s}{m+n}&&m,n\in \mathbb Z\\ \sum_k \binom{l}{m+k}\binom{s}{n+k}=\binom{l+s}{l-m+n}&&l,m,n\in\mathbb Z,l\geq0\\ \sum_k \binom{l}{m+k}\binom{s+k}{n}(-1)^k=(-1)^{l+m}\binom{s-m}{n-l}&&l,m,n\in\mathbb Z,l\geq0\\ \sum_{k\leq l}\binom{l-k}{m}\binom{s}{k-n}(-1)^k=(-1)^{l+m}\binom{s-m-1}{l-m-n}&&l,m,n\in\mathbb Z,l\geq0,m\geq0,n\geq0\\ \sum_{-q\leq k \leq l}\binom{l-k}{m}\binom{q+k}{n}=\binom{l+q+1}{m+n+1}&&m,n,l,q\in\mathbb Z,m+n\geq 0,l+q\geq0 \end{cases} \]