I have a disk with 6 equally spaced dents around the edge. The disk balances on the center point. I want to place marbles around the edge so that it stays balanced. There are four ways that this can be done:
One with two balls, one with three, one with four, and one with six. It can't be balanced with five or one.
Now the question is:
How many ways (all combinations excluding rotations and reflections) are there to balance the disk if it has 12 dents
(Imagine that my drawing skills were better and the 12 dents are all equally spaced!)
Now that there's a valid answer, I'm going to add a source with the inspiration and an interesting fact about it:
As some of the answerers noted, each answer has a complement. If it can be balanced in one configuration, then you can swap every marble for a hole and every hole for a marble and it will still be balanced.
It turns out that with $n$ holes and $k$ marbles it can be balanced if and only if both $k$ and $n-k$ can be written as the sum of prime factors of $n$.
Here's a link to the numberphile video that inspired this: Numberphile video