Question

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices $(0,0),(0,21)$ and $(21,0)$ is

• A. $133$
• B. $190$
• C. $233$
• D. $105$

Answer 1

The correct option is B $190$

$x+y=21$
The number of integral solutions to the equations are $x+y<21$, i.e., $x<21-y$
$\therefore$ Number of integral coordinates
$=19+18+....+1$
$=\frac{19(19+1)}{2}=\frac{19\times 20}{2}=190$