A ballroom contains four married couples (four men and their four wives). The gentlemen are to be paired up randomly with the ladies, one man to each woman. After one dance, the eight people are to be separated, except for the married couples that happen to be together by chance. The separated people will then be paired up again randomly for the second dance, again one man to each woman.
(a) Find the probability that exactly two married couples will be dancing together on the second dance.
(b) Suppose that exactly one married couple are dancing together on the second dance. What is the probability that no man was dancing with his wife on the first dance?