2013 Mock AIME I Problems/Problem 9
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Problem
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
.
Solution
Let be the expected value of turns it takes to short circuit
lights. Note that
,
, and in general,
Doing that calculations gives
, so
.