1. The problem statement, all variables and given/known data Twenty five men and twenty five women attend a dance competition. The com- petition rules are: There are n dance events, and each dancer must participate in all n events. In each event each man must be partnered with a woman for that dance. In each subsequent event, no man may partner with a woman with whom he was partnered during an earlier event. (b) If n=3, how many different ways are there to form new man-woman pairs for the third dance event in such a way that the competition rules are not violated? 3. The attempt at a solution for the first dance their should be 25! ways to arrange the first dance. Then for the second dance we use the derangement formula to subtract away the bad ones, but for the 3rd dance it gets quite tough. I tried doing inclusion exclusion but weird exceptions keep showing up,I tried looking for patterns but nothing seemed to work. I tried doing it for smaller groups of people. Does any one know what its called when we have n=3, would that be the 3rd permutation or is it called something else, If anyone could provide any info or where I should read more about it or what it is called. I would really appreciate it.