Question

if n is the number of triangles of different shapes (i.e., not similar ) whose angles are all integers (in degrees ), what is n/100?

Answer 1

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 = ¹⁷⁹C₂
but we have to find number of triangles of different shapes

case 1 :- if A = B = C = 60°
it is only one possible .

case 2 :- if A = B ≠ C , B = C ≠ A, C = A ≠ B
if A = 178° then, B = C = 1°
if A = 176° then, B = C = 2°
..............
........................
if A = 1° then, B = C = 89°
means, number of solution = 89
but a solutions we have to remove when A = 60° then B = C = 60°
so, number of solution = 88
similarly , 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

Answer 2

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