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Question

Mathematics Question on permutations and combinations

If in a regular polygon the number of diagonals is 5454, then the number of sides of this polygon is :

A

10

B

12

C

9

D

6

Answer

12

Explanation

Solution

Number of diagonal =54= 54
⇒n(n−3)2=54\Rightarrow \frac{n\left(n-3\right)}{2}=54
⇒n2−3n−108=0\Rightarrow n^{2}-3n-108=0
⇒n2−12n+9n−108=0\Rightarrow n^{2}-12n+9n-108 =0
⇒n(n−12)+9(n−12)=0\Rightarrow n\left(n- 12\right)+9\left(n-12\right) = 0
⇒n=12,−9\Rightarrow n=12,-9
⇒n=12(∵n≠−9)\Rightarrow n=12 \left(\because n\ne-9\right)