Combinatorial proof for two identities 
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*Does exist a combinatorial proof for the following two identities ? 


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*$\sum_{k = 0}^{n}  \binom{x+k}{k} = \binom{x+n+1}{n}$

*$\sum_{k = 0}^{n}  k\binom{n}{k} = n2^{n-1}$
I know how to derive the identites from $(1+x)^n$ , but I am searching for a combinatorial proof ? 
 A: You are choosing $n$ people from $x+n+1$ people who are lined up in a row. The right side counts the number of choices.
The left side counts the choices in another way. For any $k$ from $0$ to $n$, select all of the  first $n-k$ people (from the left), then not the next one, then choose $k$ people from the remaining $x+n+1-(n-k+1)$, that is from the remaining $x+k$.  There are $\binom{x+k}{k}$ ways to do this. The sum of the $\binom{x+k}{k}$ thus counts the choices of $n$ people from the $x+n+1$.
A: For the first identity:

$\sum_{k = 0}^{n}  \binom{x+k}{k} = \binom{x+n+1}{n}$

The left-hand side is the number of sequences of blue and yellow balls you can make such that there are exactly $x$ blue balls and at most $n$ yellow balls.
The right-hand side creates the same sequences by enumerating all sequences with $x+1$ blue balls and $n$ yellow balls, and then always removing the rightmost blue ball and all yellow balls to the right of it.
A: The second identity counts the number of nonempty sets with a distinguished representative; that is, all pairs $(x,S)$ such that $S \subseteq [n]$ with $x \in S$.


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*You can pick a set consisting of $k$ elements (in $\binom n k$ ways) and pick the representative in $k$ ways for each choice of the set. 

*You can pick the representative $x \in [n]$ first, and then choose the other members of the set in $2^{n-1}$ ways. 


Equating these two counts, we are done.
