Binomial coefficient/Sum in Pascal triangle/Fact/Proof

Proof

We have

${}{\begin{aligned}{\binom {n}{k}}+{\binom {n}{k-1}}&={\frac {n!}{(n-k)!k!}}+{\frac {n!}{(n-(k-1))!(k-1)!}}\\&={\frac {n!}{(n-k)!k!}}+{\frac {n!}{(n+1-k)!(k-1)!}}\\&={\frac {(n+1-k)\cdot n!}{(n+1-k)!k!}}+{\frac {k\cdot n!}{(n+1-k)!k!}}\\&={\frac {(n+1-k+k)\cdot n!}{(n+1-k)!k!}}\\&={\frac {(n+1)!}{(n+1-k)!k!}}\\&={\binom {n+1}{k}}.\end{aligned}}$

To fact