Puzzle #197: The Chase

There are six dogs (Bowser, Fido, Lady, Rex, Prince, and Rover) and six cats (Fluffy, Ginger, Kitty, Princess, Puss, and Tiger).

Pair each dog with a cat to chase. No dog is paired with more than one cat. No cat is paired with more than one dog.

How many unique combinations of dog-cat pairs are there? (A combination is a set of six dog-cat pairings.)

The chase is on.

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, March 15, 2017 at noon Pacific Time. I will post the answer shortly after.