$\begin{align}  & ^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n!}{r!(n-r)!}+\frac{n!}{(r-1)![n-(r-1)]!} \\ & {{\text{ }}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n!}{r!(n-r)!}+\frac{n!}{(r-1)!(n-r+1)!} \\ & {{\text{ }}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n!}{r(r-1)!(n-r)!}+\frac{n!}{(r-1)!(n-r+1)(n-r)!} \\ & {{\text{ }}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n!}{(n-r)!(r-1)!}\left[ \frac{1}{r}+\frac{1}{n-r+1} \right] \\ & {{\text{ }}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n!}{(n-r)!(r-1)!}\left[ \frac{n-r+1+r}{r(n-r+1)} \right] \\ & ^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n!}{(n-r)!(r-1)!}\left[ \frac{n+1}{r(n-r+1)} \right] \\ & ^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n!(n+1)}{(n-r+1)(n-r)!r(r-1)!} \\ & ^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}=\frac{n+1}{(n-r+1)!r!}{{=}^{n+1}}{{C}_{r}} \\\end{align}$