2016 AMC 8
Complete problem set with solutions and individual problem pages
A top hat contains red chips and green chips. Chips are drawn randomly, one at a time without replacement, until all of the reds are drawn or until both green chips are drawn. What is the probability that the reds are drawn?
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We put five chips randomly in order and then pick the chips from the left to the right. To find the number of ways to rearrange the three red chips and two green chips, we solve for . However, we notice that whenever the last chip we draw is red, we pick both green chips before we pick the last red chip. Similarly, when the last chip is green, we pick all three red chips before the last green chip. This means that the last chip must be green in all the situations that work. This means we are left with finding the number of ways to rearrange three red chips and one green chip, which is . Because a green chip will be last out of the situations, our answer is .
