Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem16

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Question number 16. A bag contains blue, green, and red marbles. It is known that if five marbles are drawn at random, then at least two will be red, and at least three will be  of the same color. How many of the marbles are blue? So let's make a list for each color. We know that there are going to be some blue marbles,  so let me write down blue here, and put one notch over here, because we are told that color exists in the bag. Likewise, there are some green marbles, so let me write down green  and put one notch here for green. And finally, there are some red marbles, and we know for sure that we will always pick two red, no matter what. Five are drawn, two are red,  so let me put two notches here for red. And so far, what I have is the total number of marbles is equal to one, two, three, four. I have four notches here. Now, we also know  that three of the marbles will be of the same color. So if I were to add one marble into our total here, I would have a total of five, and of those five, three are of the  same color. So if I were to increase the number of green by one, I would have two green, but no three of the same color, so that cannot happen. If I were to increase the number of  blue by one, I would have two blue, but no three of the same color, so I have no choice but to increase the number of red marbles by one. And so now, choosing five of them,  even if I choose a blue one and a green one, I still have three marbles remaining that are red, and this satisfies the condition that two of them are red and at least three  of the same color. So we cannot increase the number of blue or green past one, which means that exactly one of the marbles is blue. So this is our setup, and the answer is one,  just one blue marble can be in this bag.

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