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What is the number of triangles that can be formed by choosing the vertices from a set of 12 points in a plane, seven of which lie on the same straight line?
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The correct answer is:
185
To form a triangle, we need 3 points. 12 points are given.
So, ${ }^{12} \mathrm{C}_{3}$ triangles can be formed. But, given that 7 points are on a straight line. selecting 3 points from this set will not form a triangle.
So, number of triangles formed ${ }^{12} \mathrm{C}_{3}-{ }^{7} \mathrm{C}_{3}$
$=\frac{12 !}{3 ! 9 !}-\frac{7 !}{3 ! 4 !}$
$=\frac{12 \times 11 \times 10}{3 \times 2 \times 1}-\frac{7 \times 6 \times 5}{3 \times 2 \times 1}=220-35=185$
So, ${ }^{12} \mathrm{C}_{3}$ triangles can be formed. But, given that 7 points are on a straight line. selecting 3 points from this set will not form a triangle.
So, number of triangles formed ${ }^{12} \mathrm{C}_{3}-{ }^{7} \mathrm{C}_{3}$
$=\frac{12 !}{3 ! 9 !}-\frac{7 !}{3 ! 4 !}$
$=\frac{12 \times 11 \times 10}{3 \times 2 \times 1}-\frac{7 \times 6 \times 5}{3 \times 2 \times 1}=220-35=185$
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