For second one we can proof is like ,

LHS ,

$\large k\binom{n}{k}=\frac{k.n!}{k!(n-k)!}$

RHS ,

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

=$\large \frac{k.n!}{k.(k-1)!(n-k)!}$

=$\large \frac{k.n!}{k!(n-k)!}$ =LHS

so, LHS=RHS

LHS ,

$\large k\binom{n}{k}=\frac{k.n!}{k!(n-k)!}$

RHS ,

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

=$\large \frac{k.n!}{k.(k-1)!(n-k)!}$

=$\large \frac{k.n!}{k!(n-k)!}$ =LHS

so, LHS=RHS