# Eeny, meeny, miny, moe

An integer $$n$$ is chosen uniformly at random from the set $$\{512,513, \ldots, 1023\}.$$ We take $$N,$$ the binary expansion of $$n$$ and choose 2 of the digits at random. We change each of these digits, either from $$0$$ to $$1$$ or from $$1$$ to $$0,$$ to create a new binary expansion $$M,$$ which is the binary expansion for a number $$m.$$ The probability that $$m > n$$ can be expressed as $$\frac{a}{b},$$ where $$a$$ and $$b$$ are coprime positive integers. What is the value of $$a+b?$$

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