I am trying to understand the definition of a $ r- $permutation. Suppose you have $ 7 $ aligned seats and $ 7 $ different people, there are $ 7! $ Different ways to settle them.

Suppose I am trying to sit down $ 7 $ different people in $ 9 $ seats, is that when I use the formula $ P (n, r) = frac {n!} {(N-r)!} $?

My opinion is that there are two same seats (empty), so the number of distinguishable permutations is $ 9! / two! $ what is that previous formula So my question is, is $ r- $ permutation a method to discover the permutation of $ r $ objects in $ n $ boxes when $ n ge k $?

*Follow up if you have time:* If the seats formed a circle, we would divide by $ 7 $ How are the seats equivalent to the correct rotation?