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.
 A: HINT :
Using$$\binom{n}{i}=\binom{n}{n-i}$$
just add the two sums you write to get $$2S=100\binom{100}{0}+100\binom{100}{1}+\cdots+100\binom{100}{99}+100\binom{100}{100}$$
A: You can compute the sum directly, using
Note that $$r\cdot \binom nr=r\cdot \frac {n!}{r!\cdot(n-r)!}=n\cdot\frac {(n-1)!}{(r-1)!\cdot(n-r)}=n\cdot\binom {n-1}{r-1}$$
This gives you a factor $100$ you can extract from every term leaving the sum from $\binom {99}0$ to $\binom {99}{99}$. The sum of such a complete set of binomial coefficients is well known - consider $(1+1)^{99}$
A: $$\begin{align}
\sum_{i=1}^ni\binom ni
&=\sum_{i=0}^{n-1}(i+1)\binom n{i+1}
\color{lightgray}{=\sum_{i=0}^{n-1}(i+1)\frac {n(n-1)^{\underline{i}}}{(i+1)i!}=\sum_{i=0}^{n-1}n\frac {(n-1)^{\underline{i}}}{i!}}\\
&=\sum_{i=0}^{n-1}n\binom {n-1}i\\
&=n\sum_{i=0}^{n-1}\binom {n-1}i\\
&=n(1+1)^{n-1}\\
&=n\cdot 2^{n-1}
\end{align}$$
A: $\bf{My\; Solution::}$ Using $$\displaystyle (1+x)^n = \binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+..........+\binom{n}{n}x^n$$
Now Diff. both side w.r to $x\;,$ We get 
$$\displaystyle n(1+x)^{n-1} = \binom{n}{1}x+\binom{n}{2}\cdot 2x+\binom{n}{3}\cdot 3x^2+.........+\binom{n}{n}\cdot nx^{n-1}$$
Now Put $x=1$ and $n=100\;,$ We get 
$$\displaystyle 100\cdot 2^{99} = \binom{100}{1}\cdot 1+\binom{100}{2}\cdot 2+\binom{100}{3}\cdot 3+..........+\binom{100}{100}\cdot 100$$
Can anyone have any idea about combinatorial prove, If Yes Then plz explain here,
Thanks 
