Is this Gambler's fallacy? You can see the original question/quiz here

A teacher in a class of 30 students, says that he will make a random
  draw every day and the (un)lucky student who's name is drawn will be
  examined that day. 
After the 300 drawings of a whole year, the result is that the same
  student is chosen in every drawing.
The parents were upset and made allegations that the teacher hasn't
  been doing fair draws. To which he replied: "It's just the
  probabilities stupid! Every 300 draws sequence has the same chance.
  Picking the same student every day is no different than any other
  sequence of results. I'm as suspect for unfair draws as the next teacher who has drawn all students in the course of a year. Thinking otherwise is gambler's fallacy."

My mind is ready to explode. I do understand that each and every possible series of 300 drawings has the same (minuscule) probability and that 300 times the same student is just one of the many possible series. However I'm not ready to dismiss any suspicion about the results as gambler's fallacy.
Is there any way to mathematically support the claim that the draws haven't been fair or is it gambler's fallacy to think so? 
EDIT: Maybe the term 'gambler's fallacy' does not apply here in the strict definition of the term. Gambler's fallacy, requires fairness and is about predicting that 'extreme' results will 'correct themselves' (balance out) in the near future.
This question is about the reverse process: how can we evaluate the fairness of an (extreme) random outcome.
 A: The confusion comes from the contradiction of being told "every student has the same probability" and observing that only one student has been drawn.
In this case, we have observed the teacher draw the unlucky student $300$ times and draw anyone else $0$ times. Thus, what we might estimate using what has been observed that there would be a $\frac{300}{300}=100\%$ chance that the unlucky student is drawn and a $\frac0{300}=0\%$ chance that anyone else will be drawn.
The gambler's fallacy would be using the $100\%-0\%$ model to predict future outcomes rather than the known probabilities. The reason this seems so unintuitive here is that the discrepancy between the estimation and given probabilities is excessive: $\frac11$ against $\frac1{30^{300}}$ chance to draw the unlucky student every time.
Further confusing things: in the real world, a teacher would probably re-roll until they got another student or something similar. This is probably why it's a natural response to blame the teacher.

Edit (addition):
Using a chi-squared test on the observed $300$ picks of the unlucky student vs the expected $10$ picks (out of $300$) of the unlucky student, I get a $\chi^2$ value of $8420$.
Chi squared represents the discrepancy between the expected situation and the observed one, larger numbers representing higher discrepancy. It turns out that $8420$ is a ridiculous value: I could not find a calculator that would give me a significance figure for this other than $0$. Significance is the chance that we have concluded that the chance of each student being picked is not $10$ when, in reality, it is. 
Thus, we can conclude very safely here (from a statistical perspective at least) that the draw is not fair.
A: I will try to explain that yes, it is the Gambler's fallacy by rolling a die 6 times:
4, 6, 1, 2, 5, 3

No one will doubt a fairness of that dice.
But there are $6^6 = 46656$ equally probable possibilities and the one just written have the same probability as
1, 1, 1, 1, 1, 1

but for us humans it appears as something special. And by our experience special events occurs more rarely than common ones (it is something as a tautology as special events may be defined by their rare occurrences). So we expect a smaller probability of it.
But the sequence
4, 6, 1, 2, 5, 3

is a very very special one — just as special as an arbitrary other one: Try rolling a die 6 times to reach it, again and again — you have practically zero probability to succeed.

Note:
The sequences as
1, 1, 1, 1, 1, 1
3, 5, 3, 5, 3, 5
1, 2, 3, 1, 2, 3

are only very striking - because our brain is very well trained to recognize patterns.
A: In gamblers fallacy there is an extra assumption that is not guaranteed here, i.e. that the game is not rigged. For example, if you play a FAIR coin, and it has been 300 times heads, then the next flip of the coin has exactly 50% chance of being heads and 50% chance of being tails. Gamblers fallacy will be to assume that either it is more likely to be heads (because of the repetition), or more likely to be tails (so the "law of averages" works). It is 50-50. But you have to know for certain that the coin is fair. If you suspect it is not, or there is no reason to believe so, then this is no longer gamblers fallacy, and you could use some techniques (Bayesian methods come to mind) to try to gauge the chance of the next flip being heads or tails (I personally will bet heads). Of course this no longer belongs to mathematics, and we probably should not get into the hard philosophical questions of how to properly assign an a priori probability to that event.
A: The parents are being afflicted by the gamblers fallacy because they are “distributing outcomes” rather than “assigning probabilities”. In other words, they are saying that the opportunity of being selected must be distributed among all students, which means that no student can be selected more than once. Which personally I think it should it should be, but the teacher is correct in saying that it’s a gamblers fallacy because there is no exclusive selection policy in place in this example.
Another example of the gamblers fallacy explained in terms of heads or tails:
Assigning probability (what the teacher is doing)
In a 50/50 game, eg. heads or tails, each play has a 50% percent chance of being "heads" or "tails".
No play has more than more than a 50% percent chance of tails being selected.
The outcome could easily be heads for the rest of eternity.
Distributing outcomes (what the parents are doing)
In a 50/50 game there are two outcomes ("heads" or "tails")
If you distribute the outcomes to 2 plays, then one play WILL BE heads and the other play WILL BE tails.
Which means that if your first play is "heads", then your second play is guaranteed to be "tails".
So if your goal was to get tails, then your first play had a 50% chance of getting tails, 
and your second play had 100% chance of getting tails.
So the question to ask when trying to recognize a gamblers fallacy is: are they distributing outcomes?
As for a mathematical representation of fairness,
Its subjective in this case based on who’s policies you adhere to. The parents are expecting an exclusive distribution selection policy, which the teacher has not implemented and is not necessarily obliged to. The parents are expecting that no student can be selected more than once. Which means that as more students are selected, unselected students have a better chance of being selected as the remaining pool shrinks.
Fairness, in the sense of what the parents are expecting, is simply a binary value here. If that policy is not in place then the selection isn’t fair (according to the parents) and the value can be 0 (no fairness). If the policy is in place, and selected students are removed from the pool then the value is 1 (totally fair, by the parents standards). Of course the teacher is not obliged to implement that policy, so you can assert the opposite results if you go by the teachers selection standard.
