In this case, $n=12$.
We want to find the number of unique permutations of choosing 3 students from 12 students to fill the 3 positions.
The number of ways to choose the president is 12.
After choosing the president, there are 11 students left to choose the vice-president.
After choosing the president and vice-president, there are 10 students left to choose the secretary.
Therefore, the number of unique permutations is the number of ways to arrange 3 students from 12 students, which is given by the permutation formula:
$$P(n, k) = \frac{n!}{(n-k)!}$$
where $n$ is the total number of students (12) and $k$ is the number of positions to fill (3).
$$P(12, 3) = \frac{12!}{(12-3)!} = \frac{12!}{9!} = 12 \times 11 \times 10$$
$$12 \times 11 \times 10 = 1320$$
There are 1320 unique permutations of filling the 3 positions from 12 students.
Thus, there are 1320 unique permutations of these 12 students filling the 3 positions.
Final Answer: The final answer is $\boxed{1320}$