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 $M = 9\underline{87}6 \underline{54} 3210$ 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 $M$.

Let's think step by step and output the final answer within \boxed{}.
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