Question

The vertices of a regular polygon of $12$ sides are joined to form triangles. The number of triangles that do not have their sides as the sides of the polygon is?

• A. $96$
• B. $108$
• C. $112$
• D. $220$

Answer 1

The correct option is C

$112$

Explanation of the correct option:

Total number of triangles formed by $12$ sides of a polygon $={}^{12}C_3$ $...\left[{}^{n}C_r=\frac{n!}{r!(n-r)!}\right]$

Total number of triangles formed by $12$ sides of a polygon $=220$

Total number of triangles with $2$ sides as sides of the polygon$=12$

Total number of triangles with $1$ sides as sides of the polygon$=8\times 12$

Total number of triangles with $1$ sides of the polygon $=96$

Therefore total required triangles $=220-12-96$

Therefore total required triangles $=112$

Hence option C is the correct option.