Permutation: How to arrange 12 people around a table for 7

Question

I want to understand how to arrange $12$ people around a circular table with $7$ chairs. We don't care about the overflow, those people can go to another table.

I thought the way to solve the problem is that the position for the first chair is fixed, the second chair has $11$ possible options of people (since one person is already seated), the third chair has $10$ possible options, the fourth chair has $9$ possible options and so on until we get to the seventh chair which has $6$ possible options of people.

So I thought the way to solve is that this is a permutation problem $1*11*10*9*8*7*6=332640=11P6=\frac{12P7}{12}$. But my professor says the correct answer is $\frac{12P7}{7}$. I don't understand why we should divide $12P7$ by the number of chairs. Can someone explain this me?

Answer 1

I think it should be $$\binom{12}{7}6!$$