In my university we learn Set Theory prior to starting Combinatorics but they don't seem to be making a clear and explicit connection between the two. Yet it seems to me that there is in fact a very strong relation between well known combinatorial formulas like $D(n,k),c(n,k),p(n,k),n^k$ and the algebra of sets. Could someone explain it and make it explicit?
Edit
Combinations with repetitions: $D(n,k)=\binom{n+k-1}{k}=\binom{n+k-1}{n-1} = \frac{(n+k-1)!}{(n-1)!}$
Combinations without reps: $c(n,k)=\binom{n}{k}=\frac{n!}{k!(n-k)!}$
Permutations with reps: $n^k$
Permutations without reps: $p(n,k)=\frac{n!}{(n-k)!}$