Pascal's Rule

Theorem
For positive integers $$n, k \,\!$$ with $$1 \leq k \leq n \,\!$$, $${{n}\choose{k-1}} + {{n}\choose{k}} = {{n+1}\choose{k}}$$

Direct Proof
Let $$n, k \in \mathbb{N}$$ with $$1 \leq k \leq n \,\!$$.

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Q.E.D.