I have this problem here, i know that i can proof by induction but I wonder if there is a proof using combinatorials, please comment and let me know, thank you
Show that $$ \sum_{k=1}^{n} k\left(\begin{array}{l} n \\ k \end{array}\right)=n \cdot 2^{n-1} $$ for all $n \in \mathbf{Z}_{+}$.
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Take n white balls. Choose a number k at least one, Choose k balls to paint, and one of those to put a sticker on it. That’s your left hand side. This is the same as pick any of the balls to put a sticker on and paint it, then for each remaining ball choose whether to paint it. That’s the right hand side.
You want to create a team, but the number of players is not fixed (you can have at most $n$ players). You have $\binom{n}{k}$ possibilities to choose the players, and $k$ possibilities to choose a captain. So, there are $\sum_{k=1}^nk\binom{n}{k}$ different teams. An other way to compute the numbers of team you can do, is to take a captain; you have $n$ possible choice, and then to say if each player belong the the team or not. So, at the end, there are $n2^{n-1}$ possibilities.