Definition of an $r$-permutation I am trying to understand the definition of an $r$-permutation. Suppose you have $7$ aligned seats and $7$ distinct people, there are $7!$ different ways of seating them. 
Suppose that I am trying to seat $7$ distinct people in $9$ seats, is that when I use the formula $P(n,r)=\frac{n!}{(n-r)!}$? 
My thinking is there are two same seats (empty), so the number of distinguishable permutations is $9!/2!$ which is that formula above. So my question is, is $r$-permutation a method to figure out 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 $7$ as the seatings are equivalent up to rotation correct? 
 A: By r-permutation, I think u mean $P_r^n$.
You need to understand the concept behind using a formula instead of blindly applying it.
In your example, there are 9 seats and 7 people, so the total number of seating people is indeed $\frac {9!}{2!}$. You can think of it this way. For the first person, there are 9 possible seats on which he can sit. For the second person, there are 8 seats left, then there are 7 for the third and so on. The last or the seventh person will have 3 seats to choose from. Following this logic, the number of ways the people can sit is $9*8*7*6*5*4*3= \frac {9!}{2!}$. Thus, you get your required answer. This can now be generalised to r people and n seats with $n>r$. So the first has n seats to choose from, second has (n-1), third has (n-2)... until the last person(r) has (n-r+1) seats on which he can sit. So the total number of ways is $n(n-1)(n-2)...(n-r+1)$ which can be expressed as $\frac{n!}{(n-r)!}$. Hence, an r-permutation is a method to seat r people in n seats or permute r objects in n boxes where $n>r$.
For your follow up question, we would divide by 7 if the order of the seats does not matter and the seats are not numbered.
