Can someone please walk me through how to solve this?
Determine the number of graphs on $10$ vertices labeled $1, 2, \cdots, 10$ that contain exactly two out of the following four edges $e_1 = \{1, 2\}$, $e_2 = \{1, 3\}$, $e_3 = \{2, 3\}$, and $e_4 = \{1, 4\}$, and any number of other edges.