(Q). Consider three set of parallel lines, having a, b, c points respectively & no three apart from the given three are collinear
(i). if p represents number of straight lines that can pass through this system of a + b + c points excluding the three original lines then how many of the following represents correct value of p.
(1) ᵃ⁺ᵇ⁺ᶜC₂ – (ᵃC₂ + ᵇC₂ + ᶜC₂)
(2) ab + bc + ca
(3) ᵃC₁×ᵇC₁ + ᵇC₁×ᶜC1 + ᶜC1×ᵃC₁
(4) 2(ab + bc + ca)
(ii) what is the number of triangle that can be formed from the system of a + b + c points?
(iii) what is the number of quadrilaterals that can be formed from the system of a + b + c points?
Solution:-
(i). Without any restriction we can select 2 points from a + b + c points in ᵃ⁺ᵇ⁺ᶜC₂ ways but since points are collinear
∴ we will not get ᵃC₂ + ᵇC₂ + ᶜC₂ these straight lines excluding the original three lines.
Hence required number of straight lines is given by
ᵃ⁺ᵇ⁺ᶜC₂ – (ᵃC₂ + ᵇC₂ + ᶜC₂) Answer
alternatively if we select 1 point from a points and one point from b points then number of straight lines is ᵃC₁× ᵇC₁ = ab
Similarly from other two pairs we will get ᵇC₁×ᵃC₁ = ca
so total number of straight lines = ab + bc + ca Answer
(ii) Without any restriction we can select 3 points from a + b + c points in ᵃ⁺ᵇ⁺ᶜC₃ ways but since points are collinear hence we will not get ᵃC₃ + ᵇC₃ + ᶜC₃ these triangles.
Hence required number of triangle is given by
ᵃ⁺ᵇ⁺ᶜC₃ – (ᵃC₃ + ᵇC₃ + ᶜC₃) Answer
(iii). In order to from a quadrilateral we need 4 points & no three of them should be collinear so we have two cases:-
case(1):- 2 point from one straight line then 1 from each of the remaining two straight lines. In this case the number of qudrilateral is
ᵃC₂×ᵇC₁×ᶜC₁ + ᵃC₁×ᵇC₂×ᶜC₁ + ᵃC₁×ᵇC₁×ᶜC₂
case(2):- when 2 points from any of the two straight lines
ᵃC₂×ᵇC₂ + ᵇC₂×ᶜC₂ + ᶜC₂×ᵃC₂
Hence total number of such quadrilateral is :
(ᵃC₂×ᵇC₁×ᶜC₁ + ᵃC₁×ᵇC₂×ᶜC₁ + ᵃC₁×ᵇC₁×ᶜC₂) + (ᵃC₂×ᵇC₂ + ᵇC₂×ᶜC₂ + ᶜC₂×ᵃC₂) Answer