the number of integral points that lie exactly in the interior of the triangle with vertices (0,0),(0,21),(21,0) is

let the vertices of the triangle be A(21,0) ; B(0,21) and O(0,0).
thus any point in the interior of the triangle lies in first quadrant.
$therefore a>0 and b>0$
point (a,b) lies on the same side of the AB where O lies.
for (0,0) $x+y-21=-21<0$
therefore
$$\begin{gathered} a+b-21<0 \\ a+b<21 \end{gathered}$$
$$\begin{gathered} for a=1; b<21-1 i.e. b<20 ; b\in [1,19] total 19 values of b. \\ for a =2; b<21-2 i.e. b<19; b\in [1,18] total 18 integer values \\ ... \\ ... \\ similarly \\ for a=19;b<21-19 i.e. b<2 ; b=1 , 1 value \end{gathered}$$
thus number of integral points = 19+18+...........+1
$$\begin{gathered} =\frac{19*20}{2} \\ =190 integral points \end{gathered}$$
thus there are 190 integral points , which lie inside the triangle.

hope this helps you