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# Practice Paper 1 Question 8

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

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## Hints

• Hint 1
In how many ways may $$3$$ points be selected?
• Hint 2
When do $$3$$ vertices not form a triangle?
• Hint 3
Carefully consider which combinations of don't form triangles.

## Solution

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 $$\binom{16}{3}-10\binom{4}{3}-4 = 516.$$

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