well think about what relation you want, and things you know about counting it.
if people can suck their own dicks, you are interested in arbitrary bijections from k-->k, and we know the size of this set is k! (first person can be assigned any of k people, second any of k-1 remaining, etc.). if not you are counting the size of the set of bijections k-->k with no fixed points. see if there is a systematic way to remove these from the original set.