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Let \(n\) be a positive integer. Consider the sum \(x_1 y_1+x_2 y_2+\cdots+x_n y_n\) for any \(2 n\) numbers \(a_i, b_i\) taking only the values 0 and 1 . Denote by \(I(n)\) the number of \(2 n\)-tuples \(\left(x_1, x_2, \ldots, x_n, y_1, y_2, \ldots, y_n\right)\) for which this sum is odd, and by \(P(n)\) the number of those for which this sum is even. Prove that
\[
\frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1}
\]
in Mathematics by Platinum (164,920 points) | 92 views

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