The Happy Animals Kennel has $18$ cages in a row. They allocate these cages at random to $6$ dogs, $6$ cats, and $6$ pot-bellied pigs (with one animal per cage). All arrangements are equally likely. Let $A$ be the number of times in the row of cages that two animals of the same species are adjacent. For example, in the arrangement DCPCDPPPDCDPCDCCPD (where D=dog, C=cat, and P=pig), we have $A=3$. What is the expected value of $A$?

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figure c

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answer:   rotate 270° clockwise about (7, 1) and reflect across the line x = 7

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because i took test

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