Question

There are three coplanar parallel lines. If any $P$ points are taken on each of the lines, the maximum number of triangles with vertices at these points, is

• A. $^{3P}C_{3}$
• B. $P^{2} \left(P-1\right)$
• C. $P^{2} \left(4P-1\right)$
• D. $P^{2} \left(4P-3\right)$

Answer 1

Answer: (D)

$P^{2} \left(4P-3\right)$

Answer 2

Required number of ways $=\, ^{P}C_{1} \times\, ^{P}C_{1} \times\,^{P}C_{1} + \,^{P}C_{2} \times\,^{2P}C_{1} \times\, ^{3}P_{1}$ $= P^{2}\left(4P-3\right)$.