$4\times4$
points, how many triangles with their vertices on the grid can be drawn?

In how many ways may
$3$
points be selected? 
When do
$3$
vertices not form a triangle? 
Carefully consider which combinations of don’t form triangles.

Triangles are formed by choosing any
$3$
points that are not colinear. From a total of$\binom{16}{3}$
possible selected points, we exclude the combinations that form any straight lines: 10 lines pass through 4 points (4 horizontal, 4 vertical, 2 diagonals), hence
$10 \binom{4}{3}.$
 4 smaller diagonals passing through 3 points, hence
$4\cdot1.$
In total we have
$\binom{16}{3}10\binom{4}{3}4 = 516.$
 10 lines pass through 4 points (4 horizontal, 4 vertical, 2 diagonals), hence