The straight lines l_1, l_2, l_3 are parallel | vedclass

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(B) The total number of points available is $m + n + k$.The total number of ways to select $3$ points from these $m + n + k$ points is given by $^{m+n

Vedclass · Accepted Answer

$^{m+n+k}C_3 - ^mC_3 - ^nC_3 - ^kC_3$

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(B) The total number of points available is $m + n + k$.The total number of ways to select $3$ points from these $m + n + k$ points is given by $^{m+n+k}C_3$.However,a triangle cannot be formed if the $3$ selected points are collinear (i.e.,lie on the same straight line).The number of ways to select $3$ points from $m$ points on $l_1$ is $^mC_3$.The number of ways to select $3$ points from $n$ points on $l_2$ is $^nC_3$.The number of ways to select $3$ points from $k$ points on $l_3$ is $^kC_3$.Therefore,the number of triangles that can be formed is the total number of combinations minus the combinations where the points are collinear:Required number of triangles $= ^{m+n+k}C_3 - ^mC_3 - ^nC_3 - ^kC_3$.

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 $l_2$,and $k$ points on $l_3$. The maximum number of triangles formed with vertices at these points is:

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