Triangulation time

Probability Level pending

Suppose that 16 points are drawn on a plane such that exactly 7 of these points are collinear. Any set of three points which do not all belong to the 7 are noncollinear. If 3 random points are selected from 16 points, what is the probability that a triangle can be formed by joining these points?

NOTE: This problem was taken from 21st PMO Qualifying Stage.

\frac{17}{20}

\frac{15}{16}

\frac{19}{20}

\frac{63}{80}

1 solution

Nick Kent
Aug 8, 2019

We cannot form a triangle only if all three chosen points are collinear, meaning they would have to be one of the seven. There are $\begin{pmatrix} 7 \\ 3 \end{pmatrix}=35$ ways to pick three points this way. Alternatively, there are $\begin{pmatrix} 16 \\ 3 \end{pmatrix}=560$ ways to pick three points in total. SO the probability equals $1-\frac { 35 }{ 560 } =\boxed { \frac { 15 }{ 16 } }$

Triangulation time

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