Useful Combinatorial Identities

Symmetric Identity: $${n \choose k} = {n \choose n-k}$$
Decreasing Identity: $${n\choose k}=\frac{n}{k}{n-1\choose k-1}$$
Additive Identity: $${n\choose k} + {n\choose k+1} = {n+1 \choose k+1}$$
Binomial Identity: $$\sum_{k=0}^{n}{n \choose k} = {n \choose 0} + {n \choose 1} + \cdots + {n\choose n}= 2^n$$
Hockey-stick Identity: $$\sum_{k=m}^{n}{m\choose k}={m \choose m} + {m + 1\choose m} + {m + 2\choose m} +\cdots+{n \choose m} ={n+1 \choose m+1} $$
Vandermonde's Identity: $$\sum_{k=0}^{r}{m\choose k}{n\choose r-k}={m\choose 0}{n\choose r}+{m\choose 1}{n\choose r-1}+{m\choose 2}{n\choose r-2}+\cdots +{m\choose r}{n\choose 0}={m+n\choose r}$$