Graphs and Degrees · Hard problem

Let the vertices of a graph be the $10$ two-element subsets of $\{1,2,3,4,5\}$. Join two vertices by an edge exactly when the two subsets share EXACTLY one of their two elements (so $\{1,2\}$ and $\{2,3\}$ are joined, but $\{1,2\}$ and $\{3,4\}$, which share none, are not, and no subset shares two elements with a different subset). How many edges does this graph have?