Your first answer is correct.
Your second way of reasoning appears correct at first sight, but it contains many repeats amongst the Aussie selection.
Using your second method you might have a situation as follows
To select 2 Aussies, youhave 6 choices for the first pick and then 5 choices for the second pick. 6x5=30
Suppose one of these ways includes 2 Aussies named A and B. Picking A then B gives the same pair as picking B then A So divide the 30 by 2 and you have 15 ways of picking your pair of Aussies.
In one of your final selections you might have a third Aussie, C, plus any one NZer; ie A B and C and 1kiwi.
Now in another selection, reasoning by this method, you might have A and C as your 2 Aussies and then have B chosen as one of the random further picks, together with the same NZer
This is exactly the same selection as the first one.
And the duplications keep on repeating.
It could be solved by this method but you would go spare considering all of the possble duplications and eliminating them.
I hope this rather rambling reply is helpful
Cheers
mohill