The number of triangles that can be formed with the vertices of a polygon of 8 sides as their vertices if the triangle can not have any side common with the polygon.

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Solution

The correct option is D 16
Total number of triangle formed from 8 sided polygon $=^{8} C_{3} = \frac{8 \times 7 \times 6}{3 \times 2} = 56$
Number of triangle which have exactly one side common
$=^{8} C_{1} \times^{4} C_{1} = 8 \times 4 = 32$
Number of triangle which have exactly 2 sides common $= 8$
Required number of triangles with no side common with the polygon $= 56 - 40 = 16$

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