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Standard XII

Physics

Vector Addition

There are p,q...

Question

There are p,q, r points on three parallel lines $L_{1}$ , $L_{2}$ and $L_{3}$ all of which lie in one plane. Prove that the number of triangles which can be formed with vertices at their points is $^{p + q + r} C_{3} -^{P} C_{3} -^{q} C_{3} -^{r} C_{3}$

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Solution

Total number of points are p + q + r. Number of triangles is $^{p + q + r} C_{3}$ .
The points on $L_{1}$ will not form a triangle. Hence exclude $^{p} C_{3}$ and similarly exclude $^{q} C_{3}$ and $^{r} C_{3}$ for q points on $L_{2}$ and r points on $L_{3}$ .

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The straight lines $L_{1}, L_{2}, L_{3}$ are parallel and lie in the same plane. A total number of m points are taken on $L_{1}$ ,n points on $L_{2}$ , k points on $L_{3}$ .
The maximum number of triangles formed with vertices at these points are

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There are p,q, r points on three parallel lines L1 ,L2 and L3 all of which lie in one plane. Prove that the number of triangles which can be formed with vertices at their points is p+q+rC3−PC3−qC3−rC3

Total number of points are p + q + r. Number of triangles is p+q+rC3.The points on L1 will not form a triangle. Hence exclude pC3 and similarly exclude qC3 and rC3 for q points on L2 and r points on L3.

There are p,q, r points on three parallel lines $L_{1}$ , $L_{2}$ and $L_{3}$ all of which lie in one plane. Prove that the number of triangles which can be formed with vertices at their points is $^{p + q + r} C_{3} -^{P} C_{3} -^{q} C_{3} -^{r} C_{3}$

Total number of points are p + q + r. Number of triangles is $^{p + q + r} C_{3}$ .
The points on $L_{1}$ will not form a triangle. Hence exclude $^{p} C_{3}$ and similarly exclude $^{q} C_{3}$ and $^{r} C_{3}$ for q points on $L_{2}$ and r points on $L_{3}$ .