Number of integral coordinates in a given region. The number of points, having both coordinates as integers, that lie in the interior of the triangle with
vertices $(0,0) ,(0, 41$) and $(41,0)$ , is:
(1) 901   (2) 861   (3) 820   (4) 780.
I tried to find out the number of points manually, but it didn't look like the optimal method.
Thanks in advance.

Manually seems fine to me. First draw the triangle.
Consider the integral points on the line $x=1$. The point on the hypotenuse is $(1,40)$. The number of points inside the triangle are $39$, that is, $\{1,2\dots,39\}$.
Similarly number of integral points on $x=2$ are $38$, that is $\{1,2,\dots,38\}$.
So basically you have to sum up $39+38+\dots+1 = \frac{40.39}{2}=780.$
