Progressions Linear Equations

Consider a triangle drawn on the X-Y plane with its three vertices of (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

  1. 741
  2. 780
  3. 800
  4. 820

Answer

The points satisfying the equations x + y < 41, y > 0, x > 0 lie inside the triangle.

If x + y = 40, (x, y): (1, 39), (2, 38), …, (39, 1) ... (39 solutions)

If x + y = 39, (x, y): (1, 38), (2, 37), …, (38, 1) ... (38 solutions)

If x + y = 38, you get 37 solutions and so on till x + y = 2 ... (1 solution)

Number of solutions = 39 + 38 + 37 + ... + 2 + 1

Sum of Arithmetic Progression = (n/2)(a + l)

Thus, there are (39 × 40)/2 = 780 integer solutions to x + y < 41

The correct option is B.