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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.