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  4. Three vertices are chosen randomly from the seven vertices of a regular 7 -sided polygon. The probability that they form the v ertices of an isosceles triangle is

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Q. Three vertices are chosen randomly from the seven vertices of a regular $7$ -sided polygon. The probability that they form the v ertices of an isosceles triangle is

KVPYKVPY 2010

Solution:

image
We have,
Regular Heptagon from vertices ‘$A$’ There are three isosceles triangle formed
$AED, AFC, AGB$
Similarly, from other vertices also formed three isosceles triangle
$\therefore $ Required probability $=\frac{7\times3}{^{7}C_{3}}$
$=\frac{7 \times 3 \times 1 \times 2 \times3}{7 \times 6 \times 5}$
$=\frac{3}{5}$