A factorial is the number of ways to arrange $k$ things in $k$ spots.
$k * (k-1) * (k-2) * .. 1 = k!$
Permutations are a way in which a set or number of things can be ordered or arranged in which the order is important.
Combinations are a way in which a set of things can be arranged where order doesn't matter.
When we have to select $k$ things from $n$ possibilities:
Permutation - order matters - no repetition = $P(n,k)=\frac{n!}{(n-k)!}$
Permutation - order matters - yes repetition = $n^k$
Combination - order doesn't matter - no repetition = $C(n,k)= \frac{n!}{k!(n-k)!}$