ILL 1972 Problem 24

The diagonals of a convex $18$ -gon are colored in $5$ different colors, each color appearing on an equal number of diagonals. The diagonals of one color are numbered $1, 2,\cdots$ . One randomly chooses one-fifth of all the diagonals. Find the number of possibilities for which among the chosen diagonals there exist exactly $n$ pairs of diagonals of the same color and with fixed indices $i, j$ .