The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with…

$P\left(x_1, y_1\right)$ lies inside the triangle $\Rightarrow x_1, y_1\in N$

$x_1+y_1<41$

$\therefore 2\le x_1+y_1\le 40$

Number of points inside $=$ Number of solutions of the equation

$x_1+y_1=n$

$2\le n\le 40$

$x_1\ge 1, y_1\ge 1$

$\left(x_1-1\right)+\left(y_1-1\right)=\left(n-2\right)$

(Number of non - negative integral solutions of $x_1+x_2+.....+x_n=r$ is ${}^{n+r-1}{C}_r$)

Number of solutions $={}^{2+\left(n-2\right)-1}C_{n-2}$

 $={}^{n-1}{C}_{n-2}$

 $=n-1$

We have $2\le n\le 40$

$\therefore$  Number of solutions $=1+2+......+39$

 $=\frac{39\times 40}{2}$

 $=780$