If $(1+x{)}^{15}=C_0+C_{1}x+C_2{x}^2+\dots +C_{15}{x}^{15}$ then value of the expression $S=C_2+2C_3+3C_4+\dots +14C_{15}$ is

We add $-1C_0+0C_1$ to both the sides of $S$ and let
                 $S_1=S-1C_0+0C_1=S-1.$
Note that 
       $S_1=-1C_0+0C_1+1C_2+2C_3+\dots +13C_{14}+14C_{15} \left(1\right)$
Using $C_r=C_{n-r}$, we rewrite the above expression as
       $S_1=14C_0+13C_1+12C_2+\dots +0C_{14}+(-1)C_{15} \left(2\right)$
Adding $\left(1\right)$ and $\left(2\right)$, we get
$\begin{array}{l} 2S_1=13\left[C_0+C_1+C_2+\dots +C_{15}\right]=13\left(2^{15}\right)\\ \Rightarrow S_1=13\left(2^{14}\right)\Rightarrow S=13\left(2^{14}\right)+1\end{array}$