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
• Correct answer
  B

  190
• C

  233
• D

  105

Solution

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

Hence, the correct answer is option: 'B'.