The number of points, having both co-ordinates as integers, that lie in the interior of the triangles with vertices $(0, 0), (0, 41)$ and $(41, 0)$ is

901

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861

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820

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780

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Solution

The correct option is B $780$
Required point $(x, y)$ is such that it satisfies
$x + y < 41$
and $x > 0$ and $y > 0$
Number of positive integral solutions of the equation $x + y + k = 41$ will be number of integral coordinates in the bounder region.
$∴$ the total number of integral coordinates,
$=^{41 - 1} C_{3 - 1} =^{40} C_{2} = \frac{40!}{2! 38!} = \frac{40 \times 39}{2} = 20 \times 39 = 780$

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The number of points, having both co-ordinates as integers, that lie in the interior of the triangles with vertices (0,0),(0,41) and (41,0) is

The number of points, having both co-ordinates as integers, that lie in the interior of the triangles with vertices $(0, 0), (0, 41)$ and $(41, 0)$ is