2015 AMC 12A Problem 17

There are $2^8 = 256$ equally likely outcomes. Count the arrangements of standers (heads) with no two adjacent around the circle of $8$ seats, grouped by how many people stand. The number of ways to choose $k$ non-adjacent seats from a circle of $n$ is $\dfrac{n}{n-k}\dbinom{n-k}{k}.$ For $n = 8$ this gives $$1,\; 8,\; 20,\; 16,\; 2$$ for $k = 0,1,2,3,4,$ and more than $4$ standers is impossible without an adjacency. The total is $1 + 8 + 20 + 16 + 2 = 47,$ so the probability is $\dfrac{47}{256}.$ Thus, the correct answer is A.