Okay, let’s dive into terms.
Can you clarify what exactly you mean by the term "setup"? Is it the same what I called a configuration, or is it something different?
Two circles have 5 possibilities or 5 configurations or 5 variations. I agree with using configurations going forward so that we are all speaking the same language.
A setup is where you have a unique initial configuration and then two of the circles rotate through as many configurations as possible while the third circle maintains its current position.
3 circles separated would be a setup. (It would also be a configuration.) You would have 4 configurations of A and C while B stays separated, 4 for B and C while A stays separated, and 4 for A and B while C stays separated. For a total of 13 configurations.
A setup of 4 circles separated would add 12 more configurations for D and A while B and C stay separated, D and B while A and C stay separated, and D and C while A and B stayed separated.
You then need to add all the configurations of
A, B, C not separated while D remains separated, which is 12 (I think)
B, C, D while A remains separated, 12 more
C, D, A while B remains separated, 12
and D, A, B while C remains separated, 12 more.
So for the setup of 4 circles you would have 13 plus 12 (for 2 circles separated) plus 48 (for 1 circle separated).
That totals to 61, or 60 configurations plus the setup of 4 circles separated.
3 ⭕️ separated is 12+1 configurations
4 ⭕️ separated is 60+1 configurations
How many configurations would there be for a setup of 5 circles separated?