# Practice Paper 1 Question 8

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

## Related topics

## Hints

- Hint 1In how many ways may \(3\) points be selected?
- Hint 2When do \(3\) vertices not form a triangle?
- Hint 3Carefully 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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