Probability?

It depends on the preference of whoever is choosing the kittens. It is unlikely that either Brittany or her cousin will select the kittens at random. Depending on preferences, the outcome will be known with certainty. The probability of the cousin's kittens being one black and one white is then either 0 or 1. This is not the kind of problem for which probability is any use.

The number of different combinations of the 6 kittens is 6C2 which is 15. Of those combinations how many are one white and one black? The 3 white kittens can each be paired up with one of the 2 black kittens, so that means of the 15 total combinations, 6 of them are a white black combo.
So the correct probability is 6/15

6 out of 15 combinations taken 2 at a time out of an array of 6. That's a 40% chance that you get one white and one black. It's a completely different question if you ask probability of the order in which they are picked (white first, black second, or the other way around).