Yet Another Counting Problem

Atul has a set $M$, of $n$ maps. It is given that $n$ is always *even*. The set $M$ ( = $\{M_1, M_2, ..., M_n\}$) is defined as follows -  In one operation, Atul is allowed to pick any two distinct maps from the set $M$ and join point $S$ of one map to point $E$ of second map (Explanation of how maps are joint is given in the example be