Is assuming a binomial distribution appropriate when the number of possible successes is fixed? I have some data about people applying for a service - they are either successful or not successful in their application. I'm using logistic regression to investigate whether there is a relationship between being successful and some demographic variables (e.g. gender, ethnicity). 
There is a maximum to the number of people that can have access this service. So each application isn't independent - whether or not a particular person is successful can depend on who else applied. 
Is my use of logistic regression appropriate, given applications aren't independent? 
 A: In general, it's not appropriate.
The remaining question is, wether the practical impact in your application is relevant to your usecase. To assess this, you could for example perturbate the applications in the training set (let people apply in different groups than they did in originally) and check how that imacts your test error. Depending on the impact you can decide if that is acceptable.
If it's not acceptable you should probably restructure the problem.
A: As mentioned by the existing answer, it is in princple not appropriate to use logistic regression in this kind of situation. It could be that you would still get decent results, but I would go for a different approach.
I assume you will find an extensive literature on this topic, if you look into statistical models for university/college admission.
One idea would be the following: Assume that there is some kind of implicit "qualification score" that is formed by those selecting people. Let's assume that it's a continuous number (e.g. for a university entrance some kind of academic aptitude score plus bonus points for extracuricular activities) and the score everyone gets is unchanged depending on who else is applying (i.e. this would assume that there's no attempt to get the number of people with certain backgrounds balanced in any other way than giving some groups bonus points on the overall score). In that case you could look at this as observing that the successful applicants had higher qualification scores than those that were not successful. However, you do not know how the successful and unsuccessful ones were ordered - although you might learn a little bit about this if e.g. some successful applicants do accept the offer and the next most suitable ones get an offer.
