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

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Solution

Triangle formed will have vertices as
$O (0, 0), A (21, 0) and B (0, 21)$
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 ${(1, 1), (1, 2) \dots (1, 19)}$
Similarly,
when $x = 2$ , number of integral points lying inside triangle is
$18$ i.e ${(2, 1), (2, 1) \dots (2, 18)}$
$∴$ Total number of integral points lying inside triangle are $\Rightarrow 19 + 18 + 17 + \dots + 1$
$= \frac{19 \times 20}{2} = 190$

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