#### Answer

$40320$
$5040$

#### Work Step by Step

Eight dogs are selected for the final
round of a dog show to compete for best of show.
How many different ways can they line up to enter the ring?
Use permutation: $8!=40320$
The judge then randomly chooses the dogs to parade
around the ring in a circle. In how many ways can this
be done?
We assume all 8 dogs will parade around the ring.
We choose one end dog first, then the lead dog (7 choices)
then the next one (6 choices) and so on.
The total ways of arrange dogs in a circle is $7!=5040$