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 \[1 \times 1 \times 8 \times 6 \times 7\times 3\times 4\times 4\times 5\times 2 = 161280\]