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  4. There are 10 points in a plane of which no three points are collinear except 4 . Then, the number of distinct triangles that can be formed by joining these points such that atleast one of the vertices of every triangle formed is from the given 4 collinear points is

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Q. There are $10$ points in a plane of which no three points are collinear except $4 .$ Then, the number of distinct triangles that can be formed by joining these points such that atleast one of the vertices of every triangle formed is from the given $4$ collinear points is

TS EAMCET 2018

Solution:

Required number of distinct triangles
$={ }^{4} C_{1} \times{ }^{6} C_{2}+{ }^{4} C_{2} \times{ }^{6} C_{1}=4 \times \frac{6 \times 5}{2 \times 1}+\frac{4 \times 3}{2 \times 1} \times 6$
$=4 \times 3 \times 5+2 \times 3 \times 6=60+36=96$