CMO 1973 Problem 4

The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$ , $P_{0}P_{3}P_{6}$ , $P_{0}P_{6}P_{7}$ , $P_{0}P_{7}P_{8}$ , $P_{1}P_{2}P_{3}$ , $P_{3}P_{4}P_{6}$ , $P_{4}P_{5}P_{6}$ . In how many ways can these triangles be labeled with the names $\triangle_{1}$ , $\triangle_{2}$ , $\triangle_{3}$ , $\triangle_{4}$ , $\triangle_{5}$ , $\triangle_{6}$ , $\triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$ ? Justify your answer.