Question

Number of integral coordinates strictly lying inside the triangle formed by the line $x+y=21$ with coordinate axes are

Answer 1

Triangle formed will have vertices as
$O(0,0),A(21,0)\text{and}B(0,21)$
If $x=1$, Point on the hypotenuse $\equiv (1,20)$
So, number of integral points lying inside triangle when
$x=1$ is $19$ i.e $\left\{\right(1,1),(1,2)\cdots (1,19\left)\right\}$
Similarly,
when $x=2$, number of integral points lying inside triangle is
$18$ i.e $\left\{\right(2,1),(2,1)\cdots (2,18\left)\right\}$
$\therefore$ Total number of integral points lying inside triangle are $\Rightarrow 19+18+17+\cdots +1$
$=\frac{19\times 20}{2}=190$