Stirling number and cycles

Question

I was introduced stirling number of first kind [s(n,k)] as number of ways one can arrange n people around k identical tables and permuting them. But coefficent of x^k in the polynomial: x(x+1)(x+2)......(x+n-1) gives me the value of s(n,k). Can this be justified by some combinatorial interpretation (with arrangement of n people in k distinct tables and permuting them ).