Question 62
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 $1 \leq a\lt b\lt c \leq 5$, either $p(\{a, b\})\lt p(\{a, c\})\lt p(\{b, c\})$ or $p(\{b, c\})\lt p(\{a, c\})\lt p(\{a, b\})$. Find the total number of admissible orderings.