# combinatorics – \$sumlimits_{k=0}^{n} kbinom{2n}{k} = n2^{2n -1} overset{?}{iff}\$ Vandermonde’s Identity?

Does $$sumlimits_{k=0}^{r}{ {{m}choose{k}} { {n} choose {r-k} } } = {m + n choose r} overset{?}{iff}$$ $$sumlimits_{k=0}^{n} kbinom{2n}{k} = n2^{2n -1}$$ ?

If I substitute $$m = 2n$$ into Vandermonde’s Identity, then $$sumlimits_{k=0}^{r} {{m}choose{k}} {{n} choose {r-k}} = {m + n choose r} overset{m = 2n}{iff} sumlimits_{k=0}^{r} {{2n}choose{k}}{color{red}{{n} choose {r-k}}} = {3n choose r}$$.

How can I transmogrify $$color{red}{{n} choose {r-k}}$$ into $$k$$? Does this transmogrification help me succeed?