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# counting

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The Happy Valley Kennel has 4 chickens, 1 dog, and 6 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.)

Jul 14, 2022

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There are $$4!$$ ways to order the chickens

There are $$6!$$ ways to order the cats.

There is only 1 way to order the dog.

There are $$3!$$ ways to order the chickens, cats, and dogs.

So, there are $$4! \times 6! \times 3 = \color{brown}\boxed{51840}$$ ways.

Jul 14, 2022