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\[\binom{8}{2}\cdot \frac12\]?

There's a lot more than that. Remember that there are 16 edges here, not 8.

Ohh edges. Can I substitute 16 in place of 8?

Okay :|

256.

I win !!!

I used Burnside's Lemma from Group Theory using the group

I used Burnside's Lemma from Group Theory using the group D8.
D8 is also called D16 sometimes.

What's Burnside's Lemma?

I also dont know about that : )

How would you arrange 2 things in a circle at 16 places, repetition allowed

I meant repetition of the things, you can use them more than once. I am sorry :/

Ignore me. I will try to come up with concrete solution and only then reply