Question

The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the sides of the octagon is

A
24
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B
52
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C
48
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D
16
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Solution

The correct option is D 16Number of all possible triangles = Number of selections of 3 points from 8 vertices = 8C3=56 Number of triangle with one side common with octagon = 8 × 4 = 32 (Consider side A1A2. Since two points A3, A8 are adjacent, 3rd point should be chosen from remaining 4 points). Number of triangles having two sides common with octagon: All such triangles have three consecutive vertices, viz., A1A2A3, A2A3A4. ⋯A8A1A2. Number of such triangles = 8 ∴ Number of triangles with no side common = 56 - 32 - 8 = 16.

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