AoPS Wiki talk:Problem of the Day/August 16, 2011

We know ${2011\choose0}+{2011\choose1}+...+{2011\choose2011}=2^{2011}$ .

The quantity we need to find is simply:

${2011\choose0}+{2011\choose1}+...+{2011\choose2011}-{2011\choose0}-{2011\choose2011}$

$2^{2011}-{2011\choose0}-{2011\choose2011} \Rightarrow 2^{2011}-1-1=\boxed{2^{2011}-2}$ .

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