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If N is the number of triangles of different shapes whose angles are all integers (in degrees),what is N/100 ?

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2 years ago

```							It is based on permutation combination concepts. Let A, B, C are angles of triangles.we know, sum of all angles of triangle equal 180°.so, A° + B° + C° = 180° we know, according to triangle definition, A, B , C ∉ {0°, 180°} so, total number of triangles = ¹⁷⁹C₂but we have to find number of triangles of different shapescase 1 :- if A = B = C = 60° it is only one possible .case 2 :- if A = B ≠ C , B = C ≠ A, C = A ≠ Bif A = 178° then, B = C = 1° if A = 176° then, B = C = 2° ......................................if A = 1° then, B = C = 89° means, number of solution = 89but a solutions we have to remove when A = 60° then B = C = 60° so, number of solution = 88similarly , we can get 88 for A = B ≠ C an C = A ≠ B.hence, total number of solution = 88 × 3 then, total number of solutions from here = 3 × 89 - 3 = 3 × 88 = 264 case 3 :- if A ≠ B ≠ C ,then number of solutions = (¹⁷⁹C₂ - 264 -1)/3!= 2611 now, total number of triangles of different shapes = 1 + 88 + 2611 = 2700 now, n/100 = 2700/100 = 27
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one year ago
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