Prove: ${n\choose1} +2{n\choose2} +3{n\choose3} +\cdots+n{n\choose n} = n\cdot2^{n-1}$ Prove that

$$
{n\choose1} +2{n\choose2} +3{n\choose3} +\cdots+n{n\choose n} = n\cdot2^{n-1}
$$

Now I can prove this using, 

$$
{n\choose0}+{n\choose1}x+{n\choose2}x^2 + {n\choose 3}x^3 + \cdots+{n\choose n}x^n = (1+x)^{n}
$$

Differentiating both sides,

$$
{n\choose1} +2{n\choose2}x +3{n\choose3}x^2 +\cdots+n{n\choose n}x^{n-1} = n\cdot(1+x)^{n-1}
$$

Substituting $x=1$, the given identity is proved. 

My exercise specifically asks for a combinatorial proof though, by double counting.  Any help?
Consider the number of ways to choose a team (with $k\ge 1$ members) and choosing the team's captain. The left hand side counts this based on $k$ and adds the values up (for each $k$, we first choose the team members and then choose one of them as captain). Alternatively, we can first pick one person as the captain ($n$ choices) and then choose the rest of the team (zero or more members out of the remaining $n-1$ people).
$$\binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}+\cdots+n\binom{n}{n}=$$
$$=n\binom{n-1}{0}+2\frac{n}{2}\binom{n-1}{1}+3\frac{n}{3}\binom{n-1}{2}+\cdots+n\binom{n-1}{n-1}=$$
$$=n\binom{n-1}{0}+n\binom{n-1}{1}+n\binom{n-1}{2}+\cdots+n\binom{n-1}{n-1}=$$
$$=n\left(\binom{n-1}{0}+\binom{n-1}{1}+\binom{n-1}{2}+\cdots+\binom{n-1}{n-1}\right)=n2^{n-1}$$