Eleanor is going to color the sides of a regular octagon. Two sides will be red, two sides will be yellow, two sides will be green, and two sides will be blue. Furthermore, opposite sides of the regular octagon will have the same color. How many different ways can Eleanor color the sides of the ocatgon? (Two patterns are considered identical if one can be rotated to form the other.)
There are 4 ways to color the red sides, then three ways to color the yellow sides, then two ways to color the green sides, then o one way to color the blue sides. Answer = 4*3*2*1 = 24.
First, fix the orientation of the two reds in order to not overcount (this is because rotations come into play when the one color isn't fixed. Then, consider how to place the last three pairs of colors.