Interior integer points of a triangle

The interior of the triangle is bounded by $x > 0$, $y > 0$, and $x + y < 41$. Let $x + y = k$. For integer solutions inside the triangle, $k$ can range from $2$ to $40$. For a given $k$, the number of positive integer solutions $(x, y)$ is $k - 1$. Total interior points $= \sum_{k=2}^{40} (k - 1) = 1 + 2 + 3 + \dots + 39 = \frac{39 \times 40}{2} = 780$.