MHT CET · Maths · Permutation Combination
If in a regular polygon, the number of diagonals are 54 , then the number of sides of the polygon are
- A 10
- B 12
- C 9
- D 6
Answer & Solution
Correct Answer
(B) 12
Step-by-step Solution
Detailed explanation
Number of diagonals in a polygon of \(\mathrm{n}\) sides is \({ }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}\).
\(\therefore { }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}=54 \)
\( \Rightarrow \frac{\mathrm{n}(\mathrm{n}-1)}{2}-\mathrm{n}=54 \)
\( \Rightarrow \mathrm{n}^2-3 \mathrm{n}-108=0 \)
\( \Rightarrow(\mathrm{n}-12)(\mathrm{n}+9)=0\)
But, number of sides cannot be negative.
\(\therefore \mathrm{n}=12\)
\(\therefore { }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}=54 \)
\( \Rightarrow \frac{\mathrm{n}(\mathrm{n}-1)}{2}-\mathrm{n}=54 \)
\( \Rightarrow \mathrm{n}^2-3 \mathrm{n}-108=0 \)
\( \Rightarrow(\mathrm{n}-12)(\mathrm{n}+9)=0\)
But, number of sides cannot be negative.
\(\therefore \mathrm{n}=12\)
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