The Computer Laboratory

Practice Paper 1 Question 8

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

The above links are provided as is. They are not affiliated with the Climb Foundation unless otherwise specified.


  • 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.


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

If you have queries or suggestions about the content on this page or the CSAT Practice Platform then you can write to us at oi.footasc@sulp.ecitcarp. Please do not write to this address regarding general admissions or course queries.