You're always going to have at least 1 emu, chicken pair. So we make that and reduce the problem.

There are 7*5 = 35 ways to make that pair.

Now he have 4 chickens, 4 donkeys, and 6 emus to pair off.

We must pair off emus with an equal number of chickens and donkeys, otherwise we'll have a pair of the same animal left at the end.

We must pair off all the emus so we must have 3 emu/chicken pairs, and 3 emu/donkey pairs. Then we will have 1 chicken/donkey pair left.

There are 6C3 *3! * 4C3*3! ways to make the emu/chicken pairs.

Then there are 3C3*3! * 4C3*3! ways to make the emu/donkey pairs.

We then have 1 chicken and 1 donkey left that pair up.

That makes (including the 35 ways to make the first emu/chicken pair) 14515200 different arrangements.

Look at it another way.

I have 6*4 choices for the first emu/chicken pair, then 5*3, then 4*2.

I then have 3*4 choices for the first emu/donkey pair, then 2*3, then 1*2

This gives me

35 * 6*4*5*3*4*2*3*4*2*3*1*2 = 14515200