Prove that i)${ }^n{C}_{r-1}{+}^n{C}_r{=}^{n+1}{C}_r$

ii) Prove that${ }^n{C}_{r-2}+{2.}^n{C}_{r-1}+{}^n{C}_r={}^{n+2}{C}_r$

iii) For $3\le r\le n.$ Prove that${ }^{n-1}{C}_r+3{(}^{n-3}{C}_{r-1})+3{(}^{n-3}{C}_{r-2}){+}^{n-3}{C}_{r-3}$

i) L HS${ }^n{C}_{r-1}{+}^n{C}_r$

$=\frac{n!}{(n-r+1)!(r-1)!}+\frac{n!}{(n-r)!r!}$

$=n!\left[\frac{1}{(n-r+1)!(r-1)!}+\frac{1}{(n-r)!(r-1)!}\right]$

$=\frac{n!}{(n-r)!(r-1)!}\left[\frac{1}{n-r+1}+\frac{1}{r}\right]$

$=\frac{n!}{(n-r)!(r-1)!}\left[\frac{r+n-r+1}{(n-r+1)r}\right]$

$=\frac{n!}{(n-r)!(r-1)!}\left[\frac{n+1}{(n-r+1)r}\right]$

$=\frac{(n+1)!}{(n-r+1)!r!}={}^{n+1}{C}_r=RHS$

$LHS{=}^n{C}_{r-2}+{2.}^n{C}_{r-1}+{}^n{C}_r$

$=\left({ }^n{C}_{r-2}+{}^n{C}_{r-1}\right)+\left({ }^n{C}_{r-1}{+}^n{C}_r\right)$

${=}^{n+1}{C}_{r-1}+{}^{n+1}{C}_r{=}^{n+2}{C}_r$

$\therefore \boxed{{ }^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r}$

${ }^{n-3}{C}_r+3{(}^{n-3}{C}_{r-1})+3{(}^{n-3}{C}_{r-2}){+}^{n-3}{C}_{r-3}$

$=\left({ }^{n-3}{C}_r{+}^{n-3}{C}_{r-1}\right)+2{(}^{n-3}{C}_{r-1}{+}^{n-3}{C}_{r-2})+\left({ }^{n-3}{C}_{r-2}{+}^{n-3}{C}_{r-3}\right)$

$\boxed{{ }^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r}$

${=}^{n-2}{C}_r+2{(}^{n-2}{C}_{r-1}){+}^{n-2}{C}_{r-2}$

$=\left({ }^{n-2}{C}_r{+}^{n-2}{C}_{r-1}\right)+\left({ }^{n-2}{C}_{r-1}{+}^{n-2}{C}_{r-2}\right)$

${=}^{n-1}{C}_r{+}^{n-1}{C}_{r-1}{=}^n{C}_r$