Eight people are sitting around a circular table, each holding a fair

We will count how many valid standing arrangements there are (counting rotations as distinct), and divide by \(2^8 = 256\) at the end. We casework on how many people are standing.

Case 1: 0 people are standing. This yields 1 arrangement.

Case $2: 1 person is standing. This yields 8 arrangements.

Case $3: 2 people are standing. This yields (8/2) - 8 = 20 arrangements, because the two people cannot be next to each other.

Case $4: 4 people are standing. Then the people must be arranged in stand-sit-stand-sit-stand-sit-stand-sit fashion, yielding 2 possible arrangements.

More difficult is:

Case 5: 3 people are standing. First, choose the location of the first person standing (8 choices). Next, choose 2 of the remaining people in the remaining5 legal seats to stand, amounting to 6 arrangements considering that these two people cannot stand next to each other. However, we have to divide by 3 because there are 3 ways to choose the first person given any three. This yields (8*6)/3 = 16 arrangements for Case 5.

Alternate Case 5: Use complementary counting. Total number of ways to choose 3 people from 8 which is 8/3. Sub-case 1: three people are next to each other which is 8/1. Sub-case 2: two people are next to each other and the third person is not (8/1)(4/1). This yields 8/3 -8/1 - (8/1)(4/1 )= 16

Summing gives 1 + 8 + 20 + 2 + 16 = 47, and so our probability is (A) 47/256