Possibilities for the first four positions at the World Cup $2014$ 
$32$ nations participated in the World Cup $2014$.
How many possibilities were there for the order of the first four positions?

In general,
$$32 \choose 4$$
gives us the possibilities to choose $4$ nations out of $32$ nations. This doesn't include the possible positions of the $4$ nations though. For $4$ nations, there are $4!$ possibilities to rank them, so overall, we have
$${32 \choose 4} 4!$$
possibilites for the order of the first four positions.
What confuses me is that I get the same value by applying the formula
$${n! \over (n - k)!}$$
for $k \le n$.
But in our lecture notes, an example for this formula was the question: "How many possibilities are there to distribute $k$ students on $n$ places?". But in this case, wouldn't $k$ and $n$ be switched like "How many possibilites are there to distribute $32$ nations on $4$ positions?", which means that we would have $k > n$. What am I missing here?

I believe your reasoning is correct for the number of possibilities for the first four positions. 
The distinction you make between switching $k$ and $n$ seems arbitrary. We could equivalently see the problem as distributing 4 positions among 32 nations.
