Graph Coloring · Hard problem

The "house" graph has vertices $1,2,3,4,5$ with edges $12,23,34,41$ (a square) and $15,25$ (a triangular roof over edge $12$), $6$ edges in all. A painter has $4$ colors available and colors the $5$ vertices so that adjacent vertices always differ. In how many of these proper colorings is EVERY one of the $4$ colors actually used at least once?