Problem:
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
Answer Choices:
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Solution:
There are equally likely outcomes of the coin tosses. Classify the possible arrangements around the table according to the number of heads flipped. There is 1 possibility with no heads, and there are 8 possibilities with exactly one head. There are possibilities with exactly two heads, 8 of which have two adjacent heads. There are possibilities with exactly three heads, of which 8 have three adjacent heads and have exactly two adjacent heads ( 8 possibilities to place the two adjacent heads and 4 possibilities to place the third head). Finally, there are 2 possibilities using exactly four heads where no two of them are adjacent (heads and tails must alternate). There cannot be more than four heads without two of them being adjacent. Therefore there are possibilities with no adjacent heads, and the probability is .
The problems on this page are the property of the MAA's American Mathematics Competitions