Permutations And Combinations Question 341

Question: The straight lines $ {\ell_1},{\ell_2},{\ell_3} $ and parallel and lie in the same plane. A total number of m points are taken on $ {\ell_1} $ , n points on $ {\ell_2} $ . k points on $ {\ell_3} $ . The maximum number of triangles formed with vertices at these points are:

Options:

A) $ ^{m+n+k}C_3 $

B) $ ^{m+n+k}C_3{{-}^{m}}C_3{{-}^{n}}C_3{{-}^{k}}C_3 $

C) $ ^{m}C_3{{+}^{m}}C_3{{+}^{k}}C_3 $

D) None of these

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Answer:

Correct Answer: B

Solution:

  • [b] The straight line $ l_1,l_2,l_3 $ are parallel and lie in the same plane. Total number of points $ =m+n+k $ Total no, of triangles formed with vertices $ {{=}^{m+n+k}}C_3 $ By joining three given points on the same line we don?t obtain a triangle. Therefore, the max. Number of triangles $ {{=}^{m+n+k}}C_3{{-}^{m}}C_3{{-}^{n}}C_3{{-}^{k}}C_3 $