Triangles and circles but not geometry part -2

if n(side) is even , then obtuse angled triangle are - n (n\2-1) (n\2-2)/2 and n is odd then by- n (n-1)/2 (n-3)/2 so putting n=193 in 2 formula, we get - 880080 as answer

The number of triangles that can be formed by choosing any 3 vertices of 193 is 193C3 triangles. How many of these triangles are obtuse?

Let us label vertex of each obtuse triangle as A, B, C clockwise such that the angle B is obtuse. With vertex A fixed, for angle B to be obtuse, the clockwise arc from A through B to C must be less than a semicircle. In effect B and C can be anywhere among the next 96 vertices after A.

Thus the number of choices for B and C is choosing a pair of vertices out of 96 possibilities to wit 96C2= 4560.

This means that for each of 193 positions that are possible for vertex A, we have 4560 obtuse triangles, giving us 880080 obtuse triangles.