Consider a triangle drawn on the x-y plane with its 3 vertices at (0,41),(41,0) and ( 0,0). The no. of points with integer co-ordinates inside the triangle are?(excluding all points on the boundary)

The lines forming the triangle are x=0 , y = 0 and x+y = 41.
Hence all the points such that x > 0, y > 0 and x + y < 41 lie inside the triangle.
Case 1: x + y = 40 , number of possibilities = 39, x = (1,2,...39)
Case 2: x + y = 39, possibilities = 38.
....
Case 39: x + y = 2, possibilities = 1
Hence total number of points inside the triangle = 1 + 2 + .. + 39 = 780

780