Graph Coloring · Superchallenge problem

A graph $G$ has vertices $1,\ldots,6$ and edges $12,23,34,45,56,61,13,15$ ($8$ edges: the $6$-cycle plus two chords from vertex $1$). Deletion-contraction gives its chromatic polynomial as $P(G,k) = k^6 - 8k^5 + 26k^4 - 43k^3+36k^2-12k$. In general, for a graph with $m$ edges, the coefficient of $k^{n-2}$ equals $\binom{m}{2}$ minus the number of TRIANGLES. Using only this fact (not by inspecting $G$'s edges directly for triangles), how many triangles does $G$ have?