If the title says that the probability of drawing two balls, one black and one white, is 6 * 4/c (10,2) = 8/15.
If they take one black and one white at the same time, because we have calculated that the probability of one black and one white is 8/ 15 and they may be one black and one white, the probability of one black and one white is 4/15;
If you don't put it back (the order with whom doesn't affect the probability, drawing lots is the principle), then the probability is 6/ 10*4/9=4/ 15, so you can see that the modified conditions don't affect the result!
The essential reason is that you actually stipulated that A is black and B is white, which is quite in order.