Question: In a ballroom dance class, participants are divided into couples for each drill session. One partner leads and the other follows for three minutes, and then the couple switches roles for the next three minutes.
(a) Only four people show up on time. How many ways are there to pair them up?
(b) If instead six people show up on time, how many ways are there to pair them up?
(c) Assume all m people in the class arrive on time. (There are an even number of people in the class.) How many ways are there to pair them up?