jmc

Question

The Happy Valley Kennel has 4 chickens, 2 dogs, and 5 cats. (Some people in Happy Valley like to keep chickens as pets!) In how many ways can the 11 animals be placed in a row of 11 cages, such that all of the animals of each type are in adjacent cages? (Two animals of the same species are considered distinguishable.)

Accepted Answer

First we order the three groups of animals, which we can do in 3! ways. Next we order the animals within each group. There are 4! ways to arrange the group of chickens, 2! ways to arrange the group of dogs, and 5! ways to arrange the group of cats. The answer is 3! 4! 2! 5!=34,!560.