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  4. There are six vertices of a regular hexagon are chosen at random, then the possibility that the triangle with three vertices is equilateral, is equal to:

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Q. There are six vertices of a regular hexagon are chosen at random, then the possibility that the triangle with three vertices is equilateral, is equal to:

Bihar CECEBihar CECE 2004

Solution:

There are six equilateral triangle is formed in the regular hexagon.
The total number of triangle formed by the six vertices $={ }^{6} C_{3}=20$.
The total number of equilateral triangles formed $=6$
$\therefore $ Required probability $=\frac{2}{20}$
$=\frac{1}{10}$