Probability that the sum of $k$ distinct integers selected from $1, 2, \dots, n$ is divisible by $n$

Think of the numbers as integers modulo~$n$. Consider the map $A=\{a_1,a_2,\ldots,a_k\}\mapsto\{a_1+1,a_2+1,\ldots,a_k+1\}$. This adds $k$ modulo $n$ to the sum of the elements of $A$.