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Q.

Consider 6 vertices of a regular hexagon and its centre. Triangles are formed by joining any three points out of seven points. Of all possible triangles one is selected. Probability that the triangle is equilateral is:

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a

$\frac{1}{6}$

b

$\frac{1}{2}$

c

$\frac{1}{4}$

d

$\frac{1}{3}$

answer is A.

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Detailed Solution

Total number of triangles $=^{7} C_{3} - 3 = 32$
There are two equilateral triangle when $A_{0}$ is not used namely $A_{1} A_{3} A_{5} a n d A_{2} A_{4} A_{6}$ .
But when $A_{0}$
is used the number of equilateral triangle is 6.
Probability = $\frac{8}{32} = \frac{1}{4}$

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