A class in probability theory consists of 6 men and 4 women. An examination is given, and the students are ranked according to their performance. Assume that no two students obtain the same score.
(a) How many different rankings are possible?
(b) If the men are ranked just among themselves and the women just among themselves, how many different rankings are possible?
(a) Because each ranking corresponds to a particular ordered arrangement of the 10 people, the answer to this part is 10! = 3,628,800.
(b) Since there are 6! possible rankings of the men among themselves and 4! possible rankings of the women among themselves, it follows from the basic principle that there are (6!)(4!) = (720)(24) = 17,280 possible rankings in this case.