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Q.
The interior angles of a regular polygon measure $120^{\circ}$ each. The number of diagonals of the polygon is
Permutations and Combinations
Solution:
In our problem, first we need the sides of the polygon. Fact: Each interior angle of polygon of $n$ sides
$ = \frac{2n - 4}{n} \times 90^{\circ}$
$\therefore 120^{\circ} = \frac{2n - 4}{n} \times 90^{\circ}$
$\Rightarrow 6 = n$
$\therefore $ Number of diagonals
$=\,{}^nC_2 - n = \frac{n(n - 3)}{2} = \frac{6 \times 3}{2} = 9$