MathDB

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. 6740

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