I think I got some serious problem with those combinatorial proofs. Why would the following be true ($1\leq r\leq k\leq n$):
$$\sum_\limits{j=r}^{n+r-k}\binom{j-1}{r-1}\binom{n-j}{k-r} = \binom{n}{k}?$$
It doesn't make sense to me. We could choose $r$-th element of our set and choose from lesser and greater elements but this is not what this formula gives.
There are of course $\binom{n}k$ ways to choose $k$ elements from the set $[n]=\{1,\ldots,n\}$. Now let’s count them in a different way. How many ways are there to choose a set $S$ of $k$ elements of $[n]$ so that $j$ is the $r$-th smallest element of the set?
There are $j-1$ elements of $[n]$ less than $j$, so we have to choose $r-1$ of them to be the $r-1$ elements of $S$ that are less than $j$. We can do that in $\binom{j-1}{r-1}$ ways.
There will be $k-r$ elements of $S$ that are bigger than $j$, and they must come from the $n-j$ elements of $[n]$ that are bigger than $j$, so there are $\binom{n-j}{k-r}$ ways to choose them.
Thus, there are
$$\binom{j-1}{r-1}\binom{n-j}{k-r}$$
ways to choose a $k$-element subset of $[n]$ whose $r$-th smallest element is $j$. Now just sum over the possible values of $j$ to get the identity.