Red and white balls in the bags
The 4 possible cases are
1). 3 red balls,                                                   2). 2 red balls and 1 white ball,
3). 1 red ball and 2 white balls,                         4). 3 white balls.

Charlie took out 2 white balls and he could not distinguish from case 3 and case 4. It means that case 1 and case 2 were for Allan and Brian. Obviously, Allan had 3 red balls, but the label said 2 red and 1 white. Brian had 2 red and 1 white balls, but the label said 2 white and 1 red.

The left over labels are:
a). 3 red or                                                        b). 3 white

If the label Charlie had were 3 white, then he would have told immediately that he had 2 white and 1 red balls in his bag. We can be sure that the label was 3 red for Brian, 3 white for David. Because the above reason, David could tell immediately that his bag had 2 white and 1 red ball. Sure Charlie had a label of 3 red and with 3 white balls in the bag.Actually, if Charlie could think a little more he also could find out exactly the way we did it.