The straight lines l1, l2 and l3 are parallel and lie in the same plane. A total numbers 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

Question

The straight lines l1, l2 and l3 are parallel and lie in the same plane. A total numbers 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 (A)  (m+n+k) C3 (B)  (m+n+k) C3- m C3- n C3- k C3 (C)  m C3+ n C3+ k C3 (D)  m C3 × n C3 × k C3

Tardigrade Admin · Accepted Answer

Here, the total number of points are (m+n+k) which must give m+n+k C3 combinations by taking 3 points at a time, but m points lying on l1, therefore taking 3 points at a time gives m C3 combinations which produces no triangle. Similarly, n C3 and k C3 number of triangles cannot be formed. Therefore, the required number of triangles is (m+n+k) C3- m C3- n C3- k C3.

Q. The straight lines $l_{1}, l_{2}$ and $l_{3}$ are parallel and lie in the same plane. A total numbers 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

$^{(m + n + k)} C_{3}$

$^{(m + n + k)} C_{3} -^{m} C_{3} -^{n} C_{3} -^{k} C_{3}$

$^{m} C_{3} +^{n} C_{3} +^{k} C_{3}$

$^{m} C_{3} \times^{n} C_{3} \times^{k} C_{3}$

Q. The straight lines l1​,l2​ and l3​ are parallel and lie in the same plane. A total numbers 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

(m+n+k)C3​

(m+n+k)C3​−mC3​−nC3​−kC3​

mC3​+nC3​+kC3​

mC3​×nC3​×kC3​

Q. The straight lines $l_{1}, l_{2}$ and $l_{3}$ are parallel and lie in the same plane. A total numbers 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

$^{(m + n + k)} C_{3}$

$^{(m + n + k)} C_{3} -^{m} C_{3} -^{n} C_{3} -^{k} C_{3}$

$^{m} C_{3} +^{n} C_{3} +^{k} C_{3}$

$^{m} C_{3} \times^{n} C_{3} \times^{k} C_{3}$