Three vertices are chosen randomly from the eight vertices of a regular 8-sided polygon. The probability that they form the vertices of an isosceles triangle, is

Step-by-step solution

Total ways to choose 3 vertices from 8: 8 3 = 56 For isosceles triangles, fix any vertex as the apex. For vertex 1, the symmetric pairs are (2,8), (3,7), (4,6) , giving 3 isosceles triangles per apex. Since 8 is not divisible by 3, no equilateral triangles exist, so there is no overcounting. Total isosceles triangles = 8 3 = 24 P( Isosceles ) = 24 56 = 3 7