JEE Main · 2021 · 16 Mar 2021 · Shift 2 · Permutations and Combinations

Consider a rectangle

A B C D

having

5, 7, 6, 9

points in the interior of the segments

A B, C D, B C, D A

respectively. Let

α

be the number of triangles having these points from different sides as vertices and

β

be the number of quadrilaterals having these points from different sides as vertices. Then

(β - α)

is equal to :

(A)1173
(B)1890
(C)717✓ Correct
(D)795

Answer: (C)

Solution

A triangle takes one vertex from each of three different sides, and a straight line meets the boundary of a rectangle in at most two points, so no three such points are collinear and every choice gives a triangle. Summing over the four ways of choosing three sides,

α = 5 \cdot 7 \cdot 6 + 5 \cdot 7 \cdot 9 + 5 \cdot 6 \cdot 9 + 7 \cdot 6 \cdot 9 = 210 + 315 + 270 + 378 = 1173

Four points, one on each side, always form a quadrilateral, so

β = 5 \cdot 7 \cdot 6 \cdot 9 = 1890

Hence

β - α = 1890 - 1173 = 717

JEE Main · 2021 · 16 Mar 2021 · Shift 2 · Single correct option

Practise more: all JEE Main PYQs on Permutations and Combinations →

JEE Main · 2021 · 16 Mar 2021 · Shift 2 · Permutations and Combinations

Solution

More JEE Main questions on Permutations and Combinations