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 $190$

Let the vertices of the triangle be $A(21,0),B(0,21)$ and $C(0,0)$

Thus, any point in the interior of the triangle lies in first quadrant.

$\therefore a>0\&b>0$

Point (a,b) lies on the same side of the AB where O lies

For (0,0)

$x+y-21=-21<0$

therefore,

$a+b-21<0$

$a+b<21$

For $a=1;b<21-1\Rightarrow b<20:b\in [1,19]$ total 19 integral values

For $a=2;b<21-2\Rightarrow b<19:b\in [1,18]$ total 18 integral values

....

....

....

Similarly

For $a=19;b<21-19\Rightarrow b<2:b=1$ 1 integral values

Thus,

Number of integral points $=19+18+.....+1$

$=\frac{19(19+1)}{2}$

$=190$

Thus there are total $190$ integral points, which lies inside the triangle.