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+3p2  (B)  (1/2)(p3+p)  (C)  (p2/2)(5p-3)  (D)  p2(4p-3)

Tardigrade Admin · Accepted Answer

Total number of points in a plane is 3P.

∴

Maximum number of triangles

=^{3 P} C_{3} - 3.^{P} C_{3}

(here, we subtract those triangles which points are in a line)

= \frac{3 P ( 3 P - 1 ) ( 3 P - 2 )}{3 \times 2} - \frac{3 \times P ( P - 1 ) ( P - 2 )}{3 \times 2}

= \frac{P}{2} [9 P^{2} - 9 P + 2 - (P^{2} - 3 P + 2)]

= 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

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

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

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

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

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)

Total number of points in a plane is 3P. ∴ Maximum number of triangles =3PC3​−3.PC3​ (here, we subtract those triangles which points are in a line) =3×23P(3P−1)(3P−2)​−3×23×P(P−1)(P−2)​ =2P​[9P2−9P+2−(P2−3P+2)] =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

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

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

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

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