How many ways in which distinct people can get off a train I've been learning probability recently but I'm having trouble solving this question:

Suppose you have 50 people on a train and you have 4 stations you can get off at (call them Stations 1,2,3,4). If no one boards the train at any of these stations, in how many ways can the 50 people get off the train, assuming the people are distinguishable (I care who gets off where)?

If I didn't care who gets off where, then using stars and bars it would simply be 54 choose 3. But the problem for me arises in thinking when they're distinguishable. Right now my logic was for each of the 54 choose 3 ways, you can permute the groups among themselves in 4! ways. I'm pretty sure that's not right, but I don't know where to go.
We are counting the functions $f$ from a set of size $50$ to $\{1,2,3,4\}$. 
To see this, for any person $p$ let $f(p)$ be the station she gets off at. 

There are $4^{50}$ such functions.
This is a problem of the form $x_1 + \dots + x_k = n$, where $k=4$ and $n=54$. A typical ball-picking problem, which is unordered and with replacement. The answer is therefore

$$\binom{n+k-1}{k}.$$

More information here: http://mathworld.wolfram.com/BallPicking.html .