Number of circular permutations with coloring

In this problem, we need to find the number of unique ways to color a circle divided into six equal sections. We are given two red sections, one blue, one green, and one yellow. However, wait, the problem says two circles are the same if one can be rotated to match the other. Let's re-read the count: two are red, one is blue, one is green, and one is yellow. Actually, looking at the totals, five sections are colored and one is left unspecified in text, but let's check the images. Figure one has two reds, a blue, a yellow, and a green. That is five colors for six spots... wait, let's look closer. It says three sections are red. Let's re-read carefully: three sections are red, one blue, one green, and one yellow. Total of six.

Since rotations make two colorings equivalent, we can use a strategy where we fix the position of the non-red colors relative to the red ones. Let's find all the ways the three red sections can be arranged first.

Case one: The three red sections are all next to each other. We can represent the circle as a string of six slots.

In this case, we have three empty slots left for Blue, Green, and Yellow. Since the red block is fixed, these three sections can be arranged in three factorial ways.

Case two: Two red sections are adjacent, and the third one is separated by one or two slots. Let's look at the arrangement R R space R space space.

This configuration is distinct from the first because the reds are no longer all clumped. Again, we have three distinct spots left for Blue, Green, and Yellow.

Is there another way? What if the third red is two spaces away? That would be R R space space R space. Notice that this is the mirror image of the previous case, but remember only rotation is allowed for equivalence, not reflection unless specified. However, the problem says 'rotated to match'. Let's check rotation.

If we rotate R R x x R x, we cannot get R R x R x x. These are distinct under rotation. So this gives another six ways.