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

Options

A. 820
B. 780
C. 901
D. 861

Explanation

Clearly, required number of points is the total number of points in the set `S={(x,y):0ltx+ylt41,xyinN}`

Now, `0lex+yle41,x,yinN`
`implies2lex+yle40,x,yinN`
`impliesx+y=k,k=2,3,4,......40`. The number of solutions of `x+y=k" in "N " is "^(k-1)C_(2-1)=""^(k-1)C_(1)=k-1` .
Hence, required number of points `sum_(k=2)^(40)(k-1)=1+2+....+39=780`