The two most common problems in combinatorics that I have come across are: selecting some objects from a set, and distributing some objects into a set type problems.
For example, imagine you have $n$ objects that you need to choose from $r$ different types of objects, where order doesn't matter and repetition is allowed.
Another question is, you have $n$ objects that you need to put into $r$ different boxes.
In both cases, the solution is given by: $$\frac{(n+r-1)!}{n!(r-1)!}$$
I've seen this solution derived using the stars and bars method for the distribution problem.
However, is it derived in the same way for the -choosing- problem ? If so, there must be some connection between choosing and distributing type problems. In the language of combinatorics, does choose $n$ objects from $r$ types of objects, mean the same thing as distributing $n$ objects into $r$ types of bins? Or are these two fundamentally different problems that just happen to have similar mathematics working behind them ?
They're the same. Imagine you do the "choosing" by taking $n$ stickers and putting them on the chosen objects. Now you're "distributing" the stickers into $r$ "bins" corresponding to the types of objects.