$\text{ If }{C}_0,C_1,C_2,\dots ..C_n\text{ denote the coefficients of the binomial expansion }(1+x{)}^n\text{ , then the value of }{C}_1+3C_3+5C_5+\dots +\text{ is }$

$\begin{array}{l}{\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+C_3{x}^3+\mathrm{....}+C_n{x}^n\\ \text{Differentiate both sides }\\ n{\left(1+x\right)}^{n-1}=C_1+2C_{2}x+3C_3{x}^2+\mathrm{...}+n{C}_n{x}^{n-1}\\ \text{put }x=1\\ n\left(2^{n-1}\right)=C_1+2C_2+3C_3+\mathrm{...}+n{C}_n\\ \text{put }x=-1\\ 0=C_1-2C_2+3C_3+\mathrm{....}+{\left(-1\right)}^{n-1}n{C}_n\\ \text{Adding the above two equations}\\ \text{2}\left(C_1+3C_3+5C_5+\mathrm{...}\right)=n\left(2^{n-1}\right)\\ C_1+3C_3+5C_5+\mathrm{...}=n\left(2^{n-2}\right)\end{array}$