Binomial Coefficients

I understand permutations, combinations and such, but I can't seem to make sense of binomial coefficients, or at least the notation.

As an example, could someone walk me through the notation for a generic problem.. something like 100 people eligible for an award and the winner can choose 1 prize among 5 choices.

Note $$(x+y)^n=\sum_{k=0}^{n}\left(\begin{array}{cc}n\\k\end{array}\right) x^{n-k}y^{k}$$ or you can use Pascal's Triangle to get the binomial coefficients.