HMMT 2011 U36

An ordering of a set of $n$ elements is a bijective map between the set and $\{1,2, \ldots, n\}$. Call an ordering $\rho$ of the 10 unordered pairs of distinct integers from the set $\{1,2,3,4,5\}$ admissible if, for any $1 \leq a<b<c \leq 5$, either $p(\{a, b\})<p(\{a, c\})<p(\{b, c\})$ or $p(\{b, c\})<p(\{a, c\})<p(\{a, b\})$. Find the total number of admissible orderings.