Question

The number of triangles that can be formed by using the vertices of a regular polygon of $(n+3)$ sides is $220.$Then, $n$ is equal to

• A. $8$
• B. $9$
• C. $10$
• D. $11$
• E. $12$

Answer 1

The correct option is B

$9$

The explanation for the correct answer.

Solve for the number of triangles.

Since all the point are non collinear for a polygon. Therefore,

Number of triangles that can be formed using $(n+3)$ points $={}^{n+3}C_3=220$

Therefore,$\frac{(n+3)!}{3!n!}=220$

$\Rightarrow$$(n+1)(n+2)(n+3)=1320=12\times 10\times 11$

$\Rightarrow$$(n+1)(n+2)(n+3)=(9+1)(9+2)(9+3)$

$\Rightarrow$ $n=9$

Hence, option(B) is the correct answer.