The ten sides of a regular decagon are colored with five different colors, so that all five colors are used, and sides that are diametrically opposite have the same color. One possible coloring is shown below.
How many different ways are there to color the sides of the decagon? (Two colorings are considered identical if one can be rotated to form the other.)

Explanation:

Number the sides of the decagon: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, from top (currently red) clockwise.

Thus, by the multiplication or fundamental principle of counting, the number of different ways to color the decagon will be:

Notice that numbering the sides starting from other than the top side is a rotation of the decagon, which would lead to identical coloring decagons, not adding a new way to the number of ways to color the sides of the figure.

step-by-step explanation:

107 units

1 and 2

step-by-step explanation: