Number of triangles with positive area have all their vertices at points (i, j) in the coordinate plane, where i and j are integers between 1 and 5, inclusive is

Total number of ways $= 25 C_3$
Collinear points cases
Case-1 : 5 points along a line
$\begin{array}{l}\text{(1, 1) (2, 2)}\mathrm{.........}\text{(5, 5)}\\ \text{(5, 1) (4, 2)}\mathrm{.........}\text{(1, 5)}\end{array}\}$ 2 diagonals
5 Horizontal lines
$\text{(1, 1) (1, 2)}\mathrm{.........}\text{(1, 5)}$  5 Vertical lines
Total 12 sets
Number of points  ${\text{= 12. 5C}}_{\text{2}}\text{ = 120}$
Case : 2 4 points on a diagonal
$$\begin{gathered} \text{(1, 2) (2, 3)(3, 4)(4, 5) } \\ \text{(2, 1) (3, 2)(4, 3)(5, 4) } \\ \text{(4, 1) (3, 2)(2, 3)(1, 4) } \\ \text{(5, 2) (4, 3)(3, 4)(2. 5)} \end{gathered}$$
Total 4 sets
Number of points ${\text{=\hspace{0.17em}4.4C}}_{\text{3}}=16$
Case : 3 3 points on a diagonal
(2, 1) (2, 2) (1, 3)
(3, 1) (4, 2) (5, 3)
(5, 3) (4, 4) (3, 5)
(1, 3) (2, 4) (3, 5)
Total 4 sets
Number of points $={\text{4.4C}}_{\text{3}}=4$
Case : 4 Lines with slopes
a) 1/2 3 lines
b) 1/2 3 lines
c) 2    3 lines
d) –2  3 lines
Total 12 lines
Number of points $=12.3C_3=12$
Total number of ways = $25C_3-120-16-4-12=2300-152=2148$