There are 15 red, 11 blue, and 13 green cubes in a bag. all cubes are identical, except for color. How many cubes must be randomly selected to ensure that at least one pair of each color has been removed from the bag?
Hint: pigeonhole formula
A pair = 2 cubes
So if we need to ensure that at least one pair of each color has been removed, we want the worse-case scenario:
First we select red, then blue, then green (this is worse-case, because we haven't gotten a pair yet). Then we select one of the three colors (aha! one pair since all of the colors have already been selected) then we select another same color (this is worst-case because we haven't hit two pairs yet). Then we select another same color and another and another until we use up all of the red (this is worst-case because red has the most). We do that for 14 times (since there are 15 and the first red was already selected.) Now we move on to green (this is worst-case because green is the second most), we do that for 12 times (same logic). You get it. Now we select our last one which is blue (since the others are already all used up). And now you're finished. Hopefully, you can add up those cubes.