Triangles Formed by Diameters and Center

CAT 2017 Slot 1 · QA · Hard · Permutation and Combination

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Let AB,CD,EF,GH,AB, CD, EF, GH, and JKJK be five diameters of a circle with center at OO. In how many ways can three points be chosen out of A,B,C,D,E,F,G,H,J,K,A, B, C, D, E, F, G, H, J, K, and OO so as to form a triangle?

Answer

160

Explanation

Total points = 11 (10 on circle, 1 center OO). Total choices of 3 points = (113)=165\binom{11}{3} = 165. Collinear triplets cannot form a triangle. Collinear triplets are formed only by the 2 endpoints of a diameter and the center OO. There are 5 diameters, so 5 sets of collinear triplets: {A,B,O},{C,D,O},…\{A, B, O\}, \{C, D, O\}, \dots Number of triangles = 165−5=160165 - 5 = 160.

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