I've come across an identity for Stirling Numbers of the first kind: $$\sum_{i=0}^n c(n,i) = n!$$
Where $c(n,k)$ is the number of permutations on [n] with exactly k cycles. From a previous study in group theory, I recalled that a symmetric group $S_n$ contains $n!$ elements. So I wonder if there is any connection between these two? If so, can one come up with an combinatorial argument why they are the same?