I used a venn diagram to represent it.
Since no one has all three kinds, you start with putting who has 2 of different kinds (like place 4 between dogs and birds, etc)
When you're done with putting who has dogs AND cats, dogs AND birds, cats AND birds, then you subtract that from the total of each kind.
Example: If 15 total have dogs, then that includes the ones who also have dogs and birds or dogs and cats. Since 5 have dogs and cats and 4 have dogs and birds, then that's 15-9. So you have 6 who only have dogs.
Add all the numbers in the venn diagram (=24)
Since the total of people asked was 35 (your universal set) then the ones who don't have any pets is 35-24= 11.