Question

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 :

• A. 820
• B. 780
• C. 901
• D. 861

Hint:

Interior lattice points satisfy \(x\ge1\), \(y\ge1\), \(x+y\le 40\); count them by summing over \(x\) (or use Pick's theorem).

Answer 1

The Correct Option is B

Interior points \((x,y)\) of the triangle with sides \(x=0\), \(y=0\), \(x+y=41\) satisfy \(x\gt 0\), \(y\gt 0\), \(x+y\lt 41\); for integers: \(x\ge 1\), \(y\ge 1\), \(x+y\le 40\).
For a fixed \(x\) from 1 to 39, \(y\) takes values \(1,\dots,40-x\), i.e. \(40-x\) values.
Total \(= \sum_{x=1}^{39}(40-x) = 39+38+\dots+1 = \frac{39\times 40}{2} = 780\).
Answer: (2) 780