if im not mistaken... pairing to have an interval (a,b) intersecting all of the others means that for each (a_i, b_i) != (a,b) one has a_i < a, b_i > b.
so, (a,b) = (1005, 1006), a_i < 1005, b_i > 1006.
thus, suitable / total number of pairings:
\[\frac{1004!}{\binom{2010}{1005} \cdot 1005!}=...