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

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

JEE MainJEE Main 2015Permutations and Combinations

Solution:

image
$x+y<41$
$1$ to each
$x+y < 39$
$ \Rightarrow x+y \leq 38 $
$\Rightarrow x+y+z=38$
${ }^{38+3-1} C _{3-1}={ }^{40} C _{2}=\frac{40 \times 39}{2}=780 .$