<s>[INST] Find the sum of binomial coefficients 
Calculate the value of the sum 
  $$
  \sum_{i = 1}^{100} i\binom{100}{i} = 1\binom{100}{1} + 
                                       2\binom{100}{2} + 
                                       3\binom{100}{3} + 
                                       \dotsb + 
                                       100\binom{100}{100}
$$

What I have tried: 
$$\begin{align}
  S &= 0\binom{100}{0}+1\binom{100}{1}+ \dotsb +99\binom{100}{99}+100\binom{100}{100} \\ \\
    &=100\binom{100}{100}+99\binom{100}{99}+ \dotsb +1\binom{100}{1}+0\binom{100}{0}
\end{align}$$
and I'm stuck here, I don't know if it's true or not, any help will be appreciated.
 [/INST] $$\begin{align}
\sum_{i=1}^ni\binom ni
&=\sum_{i=0}^{n-1}(i+1)\binom n{i+1}
\color{lightgray}{=\sum_{i=0}^{n-1}(i+1)\frac {n(n-1)^{\underline{i}}}{(i+1)i!}=\sum_{i=0}^{n-1}n\frac {(n-1)^{\underline{i}}}{i!}}\\
&=\sum_{i=0}^{n-1}n\binom {n-1}i\\
&=n\sum_{i=0}^{n-1}\binom {n-1}i\\
&=n(1+1)^{n-1}\\
&=n\cdot 2^{n-1}
\end{align}$$
</s>