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