Combinatorial proof for two identities

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  • Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$ [duplicate]

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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 $x \in S$.

  1. 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.
  2. 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.

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.

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 $x+n+1$.