At a circular table for $100$ persons, $4$ people will shake hands with each other. How many ways are there to choose $4$ people in that group so that there are no person who shakes hands sits beside another person who shakes hands too?
I have tried something like this problem but simpler. I solve it by enumerating it manually. If I change the total number of people to $8$ and choose $3$ people that shake hands with each other, I found that the answer is $16$.
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x is the shaking hand group
Can somebody help me solving this problem. Thank you :)