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Standard XII

Mathematics

Combination with Restrictions

The straight ...

Question

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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Solution

The correct option is B

Total number of points are $m + n + k$ , the $△$ 's formed by these points = $^{m + n + k} C_{3}$

Joining 3 points on the same line gives no triangle, such $△$ 's are $^{m} C_{3} +^{n} C_{3} +^{k} C_{3}$

$∴ r e q u i r e d n u m b e r o f t r i a n g l e s o f o r m e d =^{m + n + k} C_{3} -^{m} C_{3} -^{n} C_{3} -^{k} C_{3}$

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The straight lines L1,L2,L3 are parallel and lie in the same plane. A total number of m points are taken on L1,n points on L2, k points on L3. The maximum number of triangles formed with vertices at these points are

The correct option is B Total number of points are m+n+k, the △'s formed by these points = m+n+kC3 Joining 3 points on the same line gives no triangle, such △'s are mC3+nC3+kC3 ∴ required number of triangle so formed=m+n+kC3−mC3−nC3−kC3

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The maximum number of triangles formed with vertices at these points are