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Let \(j= \dbinom{2011}{0} 3^0 + \dbinom{2011}{1} 3^1 + \dbinom{2011}{2} 3^2 + \cdots + \dbinom{2011}{1005} 3^{1005}.\) Find the remainder when \(2^{2011}-j\) is divided by \(2011^2.\)

Details and assumptions

  • This problem is not original.

  • You might use the fact that \(2011\) is prime.

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