The digits of a positive integer $$m$$ can be rearranged to form the positive integer $$n$$ such that $$m + n$$ is the 2007-digited number, each digit of which is 9. The number of such positive integers $$m$$ is ________.

Two digits add up to at most 18, so the units digits of $$m$$ and $$n$$ must add to exactly 9 with no carry, and the same holds in every column; so $$m$$ and $$n$$ both have 2007 digits and each digit $$d$$ of $$m$$ sits above $$9 - d$$ in $$n$$. Since $$n$$ is a rearrangement of $$m$$, each digit $$d$$ must occur as often as $$9 - d$$, and $$d \ne 9 - d$$, so the digits pair off and their number is even. But 2007 is odd, so no such $$m$$ exists.