p points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

Question

p points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is (A) p3+3 p2 (B) (1/2)(p3+p) (C) (p2/2)(5 p-3) (D) p2(4 p-3)

Tardigrade Admin · Accepted Answer

Total number of points in a plane is 3 p. ∴ Maximum number of triangles = 3 p C3-3 ⋅ p C3 [Here, we subtract those triangles which points are in a line] =((3 p) !/(3 p-3) ! 3 !)-3 ⋅ (p !/(p-3) ! 3 !) =(3 p(3 p-1)(3 p-2)/3 × 2)-(3 × p(p-1)(p-2)/3 × 2) =(p/2)[9 p2-9 p+2-(p2-3 p+2)] =p2(4 p -3)

Q. $p$ points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

$p^{3} + 3 p^{2}$

$\frac{1}{2} (p^{3} + p)$

$\frac{p ^{2}}{2} (5 p - 3)$

$p^{2} (4 p - 3)$

Q. p points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

p3+3p2

21​(p3+p)

2p2​(5p−3)

p2(4p−3)

Q. $p$ points are chosen on each of the three coplanar lines. The maximum number of triangles formed with vertices at these points is

$p^{3} + 3 p^{2}$

$\frac{1}{2} (p^{3} + p)$

$\frac{p ^{2}}{2} (5 p - 3)$

$p^{2} (4 p - 3)$