Answer
See explanations.
Work Step by Step
(a) $LHS=\begin{pmatrix} n\\r-1 \end{pmatrix}+\begin{pmatrix} n\\r \end{pmatrix}=\frac{n!}{(r-1)!(n-r+1)!}+\frac{n!}{r!(n-r)!}$
(b) $r!=r(r-1)!$ and $(n-r+1)!=(n-r+1)(n-r)!$, thus $r!(n-r+1)!$ is a common denominator of the fractions in part (a).
(c) $LHS=\frac{rn!}{r(r-1)!(n-r+1)!}+\frac{(n-r+1)n!}{r!(n-r+1)(n-r)!}=\frac{n!(r+n-r+1)}{r!(n-r+1)!}==\frac{(n+1)!}{r!(n-r+1)!}$.
$RHS=\begin{pmatrix} n+1\\r \end{pmatrix}=\frac{(n+1)!}{r!(n+1-r)!}=LHS$
