Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$ I want to prove the following equality:
$$\displaystyle\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$$
So I had an idea to use $((1+x)^n)'=n(1+x)^{n-1}$
So I could just use the binomial theorem and let $$x=2 \implies (n3^{n-1})$$ and then modify the sum into that one on the left side.
So I need to prove that :
$$\displaystyle\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n(1+x)^{n-1}$$ if$$x=2$$
Any help would be appreciated.
 A: From the binomial formula we have:   
$(2+x)^n = \sum_{k=0}^{n} {n \choose k} \cdot 2^{n-k} \cdot x^k$  
Let's take the derivative on both sides. We get:   
$n \cdot (2+x)^{n-1} \cdot 1 = \sum_{k=1}^{n} {n \choose k} \cdot k \cdot 2^{n-k} \cdot x^{k-1}$  
(note that the sum now starts with $k=1$ because that term which
had $k=0$ was a constant, and is now gone after taking the derivative).    
Now we put $x = 1$ and we're done with the proof.      
A: We can do this using a nice combinatorial proof. Let's say we have $n$ boxes in a row. We want to paint them all yellow, red, and blue, and one special box of choice gold. So we can first choose the special box ($n$ possibilities) and then choose for every leftover box whether we'll paint it yellow, red or blue ($3^{n-1}$ possibilities). This totals $n3^{n-1}$ possibilities.

We can also first choose the boxes that'll be yellow-ish (that is, yellow or gold). Let's say we choose $k$ of them to be so (so there's $k-1$ yellow boxes). There's $\binom{n}{k}$ possibilities for that. Then of the $k$ yellowish boxes we need to paint one gold - $k$ possibilities. For the leftover boxes, we need to choose to paint them red or blue ($2^{n-k}$ possibilities) which reaches a total of $k\binom{n}{k}2^{n-k}$ possibilities. To reach all possibilities though, we need to range $k$ from $1$ (since we need a gold box) to $n$ (one box gold and the rest yellow). So this is $\sum_{k=1}^nk\binom{n}{k}2^{n-k}$ possibilities.

Thus,
$$\sum_{k=1}^nk\binom{n}{k}2^{n-k}=n3^{n-1}$$
as we needed.
A: We can do it without derivatives using the identity $\color{red}{\binom{n+1}{k+1}={n+1\over k+1}\binom{n}{k}}$, then using the change of variable $\color{blue}{k=r+1}$ and lastly using the binomial theorem.
$$\begin{align}
S&=\sum_{k=1}^{n}k{n \choose k}2^{n-k}\color{red}=\sum_{k=1}^{n}n{n-1 \choose k-1}2^{n-k}\color{blue}=n\sum_{r=0}^{n-1}{n-1\choose r}2^{(n-1)-r}=n(1+2)^{n-1}=n3^{n-1}
\end{align}$$
