Mean for seat allocation

I am not sure you have all the information needed.
Do they leave 1 seat empty, or perhaps more than 1?
What does it mean for all seats to be filled? Does that mean there is someone in the first seat and someone in either the last or first to last seat and lots of gaps in between?
Do you know either the total number of seats or the odds of skipping more than one seat?
The mean is similar to the expected value...think of the sum over all options: (# of occupied seats) x (probability of that number).

 

So, there might be 1 or 2 seats empty, but if 3 empty seats, then not all seats are taken, since another kid could fit?
Or are there unlimited seats available, so between any two kids there might be an unlimited number of seats?
Ultimately, if you are asking to find a fraction of occupied seats, that will be # taken / # total.
If one seat available, mean fraction is 1.00.
If two seats available, mean fraction is .5
If 3 seats available, you have 2/3 full in 2 of 3 cases, and 1/3 in one of the three cases (first person sits in the 2nd seat), for a mean seat fraction of 5/9.
I am still not clear what your actual problem is, if it is one where the seats are the limiter, then it will end up being some function of seats available. If student numbers is the limiter, then it will be students/seats. If seats are unlimited, and seats between are unlimited, then you need to have some sort of probability distribution for how many seats are left between kids.

 

If the seating continues "until all seats are filled" in the row, the mean fraction of empty seats will be zero.