Consider a rectangle ABCD having 5,7,6,9 points in the interior of the line segments AB, CD , BC, DA 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 :

Question

Consider a rectangle ABCD having 5,7,6,9 points in the interior of the line segments AB, CD , BC, DA 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) 795 (B) 1173 (C) 1890 (D) 717

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/6d8057d34ae03469d064c2e8cdba17ee-.png" /> α= Number of triangles α=5 ⋅ 6 ⋅ 7+5 ⋅ 7 ⋅ 9+5 ⋅ 6 ⋅ 9+6 ⋅ 7 ⋅ 9 =210+315+270+378 =1173 β= Number of Quadrilateral β=5 ⋅ 6 ⋅ 7 ⋅ 9=1890 β-α=1890-1173=717

795

1173

1890

717

α= Number of triangles α=5⋅6⋅7+5⋅7⋅9+5⋅6⋅9+6⋅7⋅9 =210+315+270+378 =1173 β= Number of Quadrilateral β=5⋅6⋅7⋅9=1890 β−α=1890−1173=717

$α =$ Number of triangles
$α = 5 \cdot 6 \cdot 7 + 5 \cdot 7 \cdot 9 + 5 \cdot 6 \cdot 9 + 6 \cdot 7 \cdot 9$
$= 210 + 315 + 270 + 378$
$= 1173$
$β =$ Number of Quadrilateral
$β = 5 \cdot 6 \cdot 7 \cdot 9 = 1890$
$β - α = 1890 - 1173 = 717$