Tom and Sabrina just got married. It is a blended family including the pets. There are three cats and two dogs. Tom and Sabrina each had had at least one pet. How many combinations of the premarital distribution of the numbers of cats and dogs are there?

[Note: This puzzle is not asking about specific cats and dogs but the number of them. For example, if Tom had had one cat and one dog and Sabrina had had two cats and one dog, that would be one combination. Who had had which pets is not relevant here.]

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 7, 2018 at noon Pacific Time.** I will post the answer shortly after.