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Oct 12, 2011 at 8:44 comment added Gjergji Zaimi Ah, what was I thinking. This is a nice example. You've disproved the case $\binom{n+m-1}{m}\geq 2^{2n+1}$. I guess what's left is asking if the statement in the question holds for "small" $m$, for example $m\le n$.
Oct 12, 2011 at 8:23 history edited Guntram CC BY-SA 3.0
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Oct 12, 2011 at 8:11 history undeleted Guntram
Oct 12, 2011 at 8:11 history edited Guntram CC BY-SA 3.0
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Oct 12, 2011 at 7:53 history deleted Guntram
Oct 12, 2011 at 7:46 history answered Guntram CC BY-SA 3.0