Question

$C_0{C}_1+C_1{C}_2+C_{n-1}{C}_2=\frac{\left(2n\right)!}{(n-1)!(n=1)!}$

Answer 1

L. H. S. are respectively than coefficient of $x,x^2$ and $x^r$ in $\frac{1}{x^n}(1+x{)}^{2n}$ or the coefficient of $x^{n+1},x^{n+2},x^{n+r}$ in $(1+x{)}^{2n}$. These are ${}^{2n}{C}_{n+1},{}^{2n}{C}_{n+2},{}^{2n}{C}_{n+r}$ respectively

or $\frac{\left(2n\right)!}{(n-1)!(n+1)!},\frac{\left(2n\right)!}{(n-2)!(n+2)!}$ and $\frac{\left(2n\right)!}{(n-r)!(n+r)!}$