Question

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.)

Answer 1

Let the colors be red, orange, yellow, green, and blue.

First, we choose the opposite sides that are red. Since we can rotate the decagon, we can assume that the "top" and "bottom" sides are red. Going clockwise, there are 4 ways to choose the color of the next side, then 3 ways for the side after that, then 2 ways, then 1 way. Since opposite sides have the same color, all the sides of the decagon are now uniquely determined. This gives us
`$4 \cdot 3 \cdot 2 \cdot 1 = \boxed{24}$` possible colorings.

Answer 2

• 120

Step-by-step explanation:

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

• The side number one can be colored of five different colors (red, orange, blue, green, or yellow): 5

• The side number two can be colored with four different colors: 4

• The side number three can be colored with three different colors: 3

• The side number four can be colored with two different colors: 2

• The side number five can be colored with the only color left: 1

• Each of the sides six through ten can be colored with one color, the same as its opposite side: 1

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

• 5 × 4 × 3 × 2 ×1 × 1 × 1 × 1 × 1 × 1 = 120.

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.