Key binomial identities

Basic.
- $\binom{n}{k}=\frac{n!}{k!(n-k)!}$
- $\binom{n}{k}=\binom{n}{n-k}$
- $\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$ (Pascal)

Sums.
- $\sum_k\binom{n}{k}=2^n$
- $\sum_k(-1)^k\binom{n}{k}=0$ ($n\ge1$)
- $\sum_k k\binom{n}{k}=n2^{n-1}$
- $\sum_k\binom{n}{k}^2=\binom{2n}{n}$
- $\sum_k\binom{k}{r}=\binom{n+1}{r+1}$ (hockey-stick)

Vandermonde.
- $\sum_k\binom{m}{k}\binom{n}{r-k}=\binom{m+n}{r}$

Binomial theorem.
- $(x+y)^n=\sum_k\binom{n}{k}x^{n-k}y^k$

Conclusion. These form the core toolkit for combinatorial identities.