A zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the male giraffe, how many ways can he feed all the animals?
I need to count the number of possible sequences of eight feedings that alternate between male and female, starting with a male. I can use the multiplication principle to build such a sequence one feeding at a time. For the first feeding, I have only one choice: the male giraffe. For the second feeding, I have four choices: any of the four females. For the third feeding, I have three choices: any of the remaining three males, since I can't repeat the giraffe. For the fourth feeding, I have three choices: any of the remaining three females, since I can't repeat the one I just fed. For the fifth feeding, I have two choices: any of the remaining two males. For the sixth feeding, I have two choices: any of the remaining two females. For the seventh feeding, I have one choice: the last remaining male. For the eighth feeding, I have one choice: the last remaining female. So, by the multiplication principle, the total number of ways to feed all the animals is 1 * 4 * 3 * 3 * 2 * 2 * 1 * 1 = 144. 