A circle is divided into three sectors: $S_1,S_2,S_3$. The central angle of each sector, in degrees, is a nonnegative integer. What is the number of distinct circles that can be made if:
- $S_1$ is colored black, $S_2$ is colored blue and $S_3$ is colored yellow.
- The sectors all have the same color. Circles that are the same under rotation are considered same.
- The sectors all have the same color. Circles that are the same under rotation or reflection are considered same.
Let $a,b,c$ be the degree measure of the sectors.
In Case 1 since the sectors are colored, they are distinguishable, and we need the ordered triples that are integer solutions to the Diophantine equation below (done using stars and bars with 360 objects and 2 dividers)
$$a+b+c=360\implies {362\choose2} \text{solutions}$$
Case 3 requires unordered triples that are solutions to the same equations above. I recognize that
- Case $A$: $a,b,c$ are distinct, the number of solutions needs to divided by $6$
- Case $B$: Exactly two of $a,b,c$ are distinct, the number of solutions needs to be divided by $3$
- Case $C$: All of $a,b,c$ are the same, the number of solutions remains unchanged
However, I am not sure how to break down the $362\choose2$ solutions into the three cases above.
Case 2 does not include reflections. How does that increase the number of solutions compared to Case $3$. I am not sure how to do this.