Question

Three of six vertices of a regular hexagon are chosen at random. The probability that triangle with three vertices is equilateral, is equal to

• A. $\frac{1}{5}$
• B. $\frac{2}{5}$
• C. $\frac{1}{10}$
• D. $\frac{1}{20}$

Answer 1

Answer: (C)

$\frac{1}{10}$

Let $T$ represent the equilateral triangles
$n\left(T\right)=$ No. if equilateral triangle, with six vertices $ABCDEF$ of regular hexagon, are only two namely $△ACE;△BDF$
and $n\left(S\right)=$ Total no. of triangles with six vertices are $=={{}^{6}C}_3=20$
$P\left(T\right)=\frac{n\left(T\right)}{n\left(S\right)}=\frac{2}{20}=\frac{1}{10}$