Three distinct vertices of a regular 25-sided polygon are selected at random. What is the probability that the triangle formed by these three vertices contains the center of the polygon?
-649 The total number of ways to choose 3 distinct vertices from a 25-sided polygon is C(25,3) = 2300.
A triangle formed by 3 vertices of a regular polygon contains the center of the polygon if and only if the 3 vertices do not form an isosceles triangle. There are C(24,2)=276 ways to choose 2 vertices that form an isosceles triangle, so there are 2300−276=2024 ways to choose 3 vertices that do not form an isosceles triangle.
Therefore, the probability that a triangle formed by 3 randomly chosen vertices of a regular 25-sided polygon contains the center of the polygon is 2024/2300 = 22/25.
