A particular firm has six vacancies to fill from (C) 15 applicants. Calculate the number of ways in which these vacancies could be filled if there are no restrictions. The firm decides that three of the six vacancies shall be filled by women and three by men. The applicants consist of seven women and eight men. Calculate the number of ways in which the six vacancies could be filled under these conditions. One of the seven women is the wife of one of the eight men. Calculate the number of ways in which three women and three men could fill the six vacancies, given that both the wife and her husband are among those appointed.

Of all the possible selections of three women and three men, one is picked at random. Calculate the probability that this selection includes

(a) both the wife and her husband,

(b) either the wife or her husband, but not both. (C)