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Plug in an arbitrary $X$ into the formula. Look at each summand for each particular $Y \in \{0,1\}^N$ If $Y \neq X$, then there must be at least one index $i$ such that $X_k \neq Y_k$. But both $Y_k$ and $X_k$ are only either $0$ or $1$. So if they are not equal, then the difference must be either $+1$ or $-1$. Square that and you get $1$ if they are ...


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