If four cats and five mice enter a race

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If four cats and five mice enter a race, how many ways can they finish with a mouse placing first, second, and third?

I was about to say that the answer would be P 9, 5 but not sure...
 
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Seems to me like the answer would be 6! = 720. But I might be talking out of my ass.
 
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5*4*3*6!

In first place, there are only 5 choices, one of the mice. In second place, there are only 4, since one of the mice is already chosen. In third, there're 3 choices. In the remaining, all choices are open.

cookiemonster

Edit: Noticed a weird word in there that didn't belong.
 
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are the mice distinguishable...? it will influence the answer.
 

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