How many ways are there to sit 4 people from a group of 10 people around a circular table where two sittings are considered

Question

How many ways are there to sit 4 people from a group of 10 people around a circular table where two sittings are considered the same when everyone has the same immediate left and immediate right neighbor?

Wouldn't the answer just be $4*10 = 40$? or would it be $4! = 24$

Answer 1

First you need to count how many ways you can choose $4$ people from $10$:

That is exactly $\frac{10!}{4!*(10-4)!}$ and then you need to multiply this with $3!$, the possibilities to sit in a circle for that $4$ people, since for $n$ people, the possibilties to sit around in a circle are $(n-1)!$

The final answer: $3! * \frac{10!}{4!*(10-4)!} = 1260$.

Answer 2

First choose $4$ of the $10$ people to be seated. [$10\choose 4$ ways.]

Then seat these people in the four seats. [$4!$ ways.]

Then divide by $4$ since you overcount by a factor of $4$ due to rotations being considered the same.