# Graphs containing specific edges

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.

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Do you need the graphs connected? –  EuYu Nov 5 '12 at 4:43
no, the graph does not need to be connected –  entreband Nov 5 '12 at 4:52

1. How many edges are possible on $10$ vertices?
2. How many possible edges are there besides the $4$ edges $e_1,e_2,e_3$, and $e_4$? Call this number $m$.
3. In how many ways can you choose $2$ of the $4$ edges $e_1,e_2,e_3$, and $e_4$?
4. In how many ways can you choose an arbitrary subset of the $m$ edges other than these four?
5. Each of the pairs of edges from $\{e_1,e_2,e_3,e_4\}$ can be combined with any subset of the other $m$ edges; how many different graphs does this produce?