# Probability – Definition of a permutation \$ r- \$

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?