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GATE Overflow Test Series | Mixed Subjects | Test 1 | Question: 12

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Using three points we can form a triangle. First we select $3$ points from the pool of $15$ points, the number of ways  $=\:^{15}C_{3} = 455.$

But $6$ points are collinear, meaning these $6$ points do not make any triangle. So, number of ways of not forming a triangle $ =\:^{6}C_{3} = 20.$

Therefore, total number of triangles possible
$ = 455-20 = 435$.

$\textbf{General formula:}$ $^{n}C_{3} -\: ^{r}C_{3}$

So, the correct answer is $(D)$.
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