How many triangles can be obtained by joining 12 points, five of … - Vidyadip

Question

How many triangles can be obtained by joining 12 points, five of which are collinear?

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Answer

We have,
Since 5 out of 12 points are collinear, So the number of triangle will be,
$={^{12}{\text{C}}}_{\text{3}}-{^{5}{\text{C}}}_{\text{3}}$
$=\frac{12!}{3!9!}-\frac{5!}{3!2!}$
$=\frac{12\times11\times10}{3\times2}-\frac{5\times4}{2}$
$=220-10$
$=210$

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