How to select 3 objects from n different objects arranged in a circle such that no two objects are consecutive?

Select a fixed point on the circle to start with, and read out the choices in clockwise order.

In order to find the number of outcomes where the first position is not chosen, you need to find some combination of 3 times "no, then yes" and n−6 times "no" that is, (ⁿ⁻³ C ₃) different outcomes.
Now for the number of outcomes where the first position is chosen. We can make every such outcome by taking one the the ones from before where we started with "no, then yes", and then rotating the entire pattern one position counterclockwise. This gives us (ⁿ⁻⁴ C ₂) different outcomes.