Binomial Identities

In the setting of Finite Combinatorics: Notation, let $m,n,k\in\mathbb{N}_{0}$.

(i) Row sum.

$$\sum_{j=0}^{n}\binom{n}{j}=2^{n}.$$

(ii) Alternating row sum. In $\mathbb{R}$,

$$\sum_{j=0}^{n+1}(-1)^{j}\binom{n+1}{j}=0.$$

(iii) Absorption.

$$(k+1)\binom{n+1}{k+1}=(n+1)\binom{n}{k}.$$

(iv) Vandermonde's identity.

$$\sum_{j=0}^{k}\binom{m}{j}\binom{n}{k-j}=\binom{m+n}{k}.$$

(v) Hockey-stick identity.

$$\sum_{i=0}^{n}\binom{i}{k}=\binom{n+1}{k+1}.$$

(vi) Hockey-stick identity, shifted form.

$$\sum_{s=0}^{n}\binom{s+k}{k}=\binom{n+k+1}{k+1}.$$

(vii) Weighted row sum.

$$\sum_{j=0}^{n+1}j\binom{n+1}{j}=(n+1)\,2^{n}.$$