The binomial coefficient basically provides the number of ways to choose a set of $k$ from $n$ sets. To me, it can be considered the number of unique ways to pick $k$ amount of "cards" from a deck of $n$.
However, I don't see why this provides the coefficient of the polynomial. Why does the number of unique ways to pick $k$ numbers from a set of $n$ provide the coefficient of the polynomial?
I already looked at the combinatorial explanation offered by Wikipedia, however I do not understand it.