What you've done here is to choose a pair of animals first (and you're right there are 144 ways of doing this), and then to allocate for them a month, a time and an element.
The problem with this is that it assumes that each animal in the pair that has been chosen, has the same month, time and element. I presume that you didn't intend this (although you haven't said so), in which case you need to say:
- The male animal can be allocated a month, time and element in 12 x12 x 5 ways.
- The female animal can be allocated a month, time and element in 12 x 12 x 5 ways.
So once the pair has been chosen (in 12 x 12 ways) we allocate each member of the pair its month, time and element in (12 x 12 x 5) x (12 x 12 x 5) ways.
So the total number of choices altogether is (12 x 12) x (12 x 12 x 5) x (12 x 12 x 5) = 74 649 600