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  4. In a regular 15 -sided polygon with all its diagonals drawn, a diagonal is chosen at random. The probability that it is either a shortest diagonal nor a longest diagonal is

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Q. In a regular $15$ -sided polygon with all its diagonals drawn, a diagonal is chosen at random. The probability that it is either a shortest diagonal nor a longest diagonal is

KVPYKVPY 2020

Solution:

Total diagonals $={ }^{15} C _{2}-15=90$
Shortest diagonal = Diagonal connecting
$\left( A _{1} A _{3}, A _{2} A _{4}, \ldots\right)$
$=15$
image
longest diagonal = Diagonal connecting
$\left(A_{1} A_{8}, A_{1} A_{9}, \ldots\right)$
$=15$
Required probability $=\frac{90-15-15}{90}$
$=\frac{60}{90}=\frac{2}{3}$