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

Step-by-step solution

P x 1 ,   y 1 lies inside the triangle ⇒   x 1 ,   y 1 ∈ N x 1 + y 1 < 4 1 ∴    2 ≤ x 1 + y 1 ≤ 4 0 Number of points inside = Number of solutions of the equation x 1 + y 1 = n 2 ≤ n ≤ 4 0 x 1 ≥ 1 ,   y 1 ≥ 1 x 1 - 1 + y 1 - 1 = n - 2 (Number of non - negative integral solutions of x 1 + x 2 + . .... + x n = r is n + r - 1 C r ) Number of solutions = C n - 2 2 + n - 2 - 1 = n - 1 C n - 2 = n - 1 We have 2 ≤ n ≤ 4