Consider the $$10$$-digit number $$M=9876543210$$. We obtain a new $$10$$-digit number from $$M$$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $$M$$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $$M=9876543210$$, by interchanging the $$2$$ underlined pairs, and keeping the others in their places, we get $$M_{1}=9786453210$$. Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from $$M$$.