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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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