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 $$n, k \in \N$$ with $$1 \leq k \leq n \,\!$$.

$$ $$ $$ $$ $$ $$ $$ $$