${{\mathrm{C}}_1}^2+2.{{\mathrm{C}}_2}^2+3.{{\mathrm{C}}_3}^2+\dots \dots +\mathrm{n}.{{\mathrm{C}}_{\mathrm{n}}}^2=$

${{\mathrm{C}}_1}^2+2 {{\mathrm{C}}_2}^2+3 {{\mathrm{C}}_3}^2+....+\mathrm{n} {{\mathrm{C}}_{\mathrm{n}}}^2$  is equal $\frac{\mathrm{n}}{2}.{}^{2\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}$

$(1+x{)}^n={\mathrm{C}}_0+{\mathrm{C}}_{1}x+\dots \dots +{\mathrm{C}}_n{x}^n$

Differentiate with respect to x

$n(1+x{)}^{n-1}={\mathrm{C}}_1+2{\mathrm{C}}_{2}x+3{\mathrm{C}}_3{x}^2+\dots .+n{\mathrm{C}}_n\cdot x^{n-1}$

$\text{ and }(x+1{)}^n={\mathrm{C}}_0{x}^n+{\mathrm{C}}_1{x}^{n-1}+\dots ..+{\mathrm{C}}_n$

Multiplying above two equations

$\begin{array}{r}n(1+x{)}^{n-1}\cdot (x+1{)}^n=\left({\mathrm{C}}_1+2{\mathrm{C}}_{2}x+3{\mathrm{C}}_3{x}^2+\dots .{+}^n{\mathrm{C}}_n\cdot x^{n-1}\right)\\ \left({\mathrm{C}}_0{x}^n+{\mathrm{C}}_1{x}^{n-1}+\dots .+{\mathrm{C}}_n\right)\end{array}$

$\text{ And equate coefficient of }{x}^{n-1}\text{ on both sides, we get }$

$\Rightarrow C_1^2+2{\mathrm{C}}_2^2+3{\mathrm{C}}_3^2+\dots +n{\mathrm{C}}_n^2=n{\cdot }^{2n-1}{\mathrm{C}}_{n-1}$