# Ten for ten

Let $$S$$ be the set of all $$10$$-digit integers that can be composed from the integers $$1,2, ...., 9,$$ (repetition is allowed). Let $$A$$ be the product of the $$10$$ digits of an element $$N$$ of $$S$$ that has been chosen uniformly and at random.

If $$P$$ is the probability that $$A$$ is divisible by $$10$$, then find $$\lfloor 1000P \rfloor$$.

Details and Assumptions:

As an explicit example: if $$N = 1186734452$$, then the value of $$A$$ associated with $$N$$ is $1 \times 1 \times 8 \times 6 \times 7\times 3\times 4\times 4\times 5\times 2 = 161280$

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