Probable Near Pandigital Poser (Posted on 2010-01-31)

Determine the probability that a 15-digit base ten positive integer with no leading zero contains each of the digits from 0 to 9 at least once, but at most twice.

As a bonus, determine the probability that a 15-digit base ten positive integer having no zero digit contains each of the digits from 1 to 9 at least once, but at most twice.