2013 Mock AIME I Problems/Problem 9
In a magic circuit, there are six lights in a series, and if one of the lights short circuit, then all lights after it will short circuit as well, without affecting the lights before it. Once a turn, a random light that isn’t already short circuited is short circuited. If is the expected number of turns it takes to short circuit all of the lights, find .
Let be the expected value of turns it takes to short circuit lights. Note that , , and in general, Doing that calculations gives , so .