Final answer:
The number of points with integer coordinates strictly inside the triangle is found by adding integers from 1 to 39, which forms an arithmetic series, giving us a total of 780 points. Therefore the correct answer is option A.
Step-by-step explanation:
The question asks for the number of points with integer coordinates inside a triangle on the X-Y plane with vertices at (41, 0), (0, 41), and (0, 0). To solve this, observe that the triangle formed is right-angled with its right angle at the origin, and the points with integer coordinates that lie inside the triangle are those points (x, y) for which both x and y are integers, x > 0, y > 0, and x + y < 41.
To find the total number of such points, we can count the points with integer coordinates on each horizontal line y = 1, y = 2, ..., y = 39 (excluding y = 40 because then x = 1 which lies on the boundary). For y = 1, there are 39 points (x = 1 to 39), for y = 2, there are 38 points (x = 1 to 38), and so on, until y = 39, which has 1 point (x = 1). Summing these gives a total of 1+2+...+39 which is a sum of an arithmetic series.
The sum of the first n natural numbers is given by n(n+1)/2, so the sum of the first 39 numbers is 39*40/2 = 780.