The question is explicitly: "In how many different orders can $n$ cars line up in a gas station of 4 parallel gas pumps?" No more information is provided. I initially assumed that both the pumps and cars are distinguishable.
In these cases, different orders count towards the total number of possible outcomes, and repetition is not allowed since the same car can't be parked in two locations. Would that mean the solution would take the form $P(n,r)$? But that would reduce to $n!$ which is just the number of different arrangements the cars can take in a single line.
Would I just multiply by $4$ to get the number of arrangements if they're distributed across $4$ gas stations?
Instinctively it should look more like
(n - # of cars @ 2,3,4)! + (n - # of cars @ 1,2,4)! + (n - # of cars at 1,2,4)! + (n-# of cars at 1,2,3)!
How can I express this purely in terms of n?