Actually, many of the animals have more than one combinations, as long as you choose the right gene. For example, you can choose elephant or sea devil as [giant] gene, both will work.
There are 58,392 working combinations to be exact
so it should actually be pretty easy to make the 525 animals just by guesswork.
As SteveG_UK pointed out, the order of the genes does matter.
example:
chicken's genes = Bird, Dumb, Domestic.
rat + chicken [dumb] = mole
rat + tweety [domestic] = hamster
so you see I cannot make hamster with rat + chicken [domestic] as Dumb comes first in chicken's gene list, so rat + chicken [dumb] = mole. but I can make hamster with rat + tweety
bit more info:
Each animal has a parent animal or gene. ant + chicken [domestic] = pharaoh ant. so pharaoh ant's parent is ant.
Each animal also has a "Main Gene", this is the gene required to make the animal. ant + chicken [domestic] = pharaoh ant. so the pharaoh ant's "Main Gene" is domestic.
so when you combine two animals (parent animal and "Gene donor") it goes through the "Gene donor"s genes in order and looks for an animal that has Parent and Main Gene matching your current combination. if theres a match, you just made a new animal! hooray!
also some animals have a gene as the father (for example shrimp's father is "fish" but you can make it with "anchovy" which has "fish" as a gene).
this stuff can all be worked out using the hints button, info button, and extensive use of excel
just to confuse you even more..