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JEE Main 202116 MarchEvening Shift
Exam
JEE Main
Year
2021
Date
16 March
Shift
Evening Shift
Subject
Mathematics
Chapter
Permutation and Combination

JEE Main 2021 · 16 March · Evening Shift · Mathematics · Permutation and Combination

Consider a rectangle ABCD having 5, 6, 7, 9 points in the interior...

Consider a rectangle ABCDABCD having 5,6,7,95, 6, 7, 9 points in the interior of the line segments AB,BC,CD,DAAB, BC, CD, DA respectively. Let α\alpha be the number of triangles having these points from different sides as vertices and β\beta be the number of quadrilaterals having these points from different sides as vertices. Then (β−α)\left(\beta - \alpha\right) is equal toConsider a rectangle ABCD having 5, 6, 7, 9 points in the interior of the line segments AB, BC, CD, DA respectively.

Options

  1. A795795
  2. B11731173
  3. C18901890
  4. D717717

Correct answer

D. 717717

Step-by-step solution

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

α=\alpha = Number of triangles

α=5⋅6⋅7+5⋅7⋅9+5⋅6⋅9+6⋅7⋅9\alpha = 5 \cdot 6 \cdot 7 + 5 \cdot 7 \cdot 9 + 5 \cdot 6 \cdot 9 + 6 \cdot 7 \cdot 9

=210+315+270+378= 210 + 315 + 270 + 378

=1173= 1173

β=\beta = Number of Quadrilateral

β=5⋅6⋅7⋅9=1890\beta = 5 \cdot 6 \cdot 7 \cdot 9 = 1890

β−α=1890−1173=717\beta - \alpha = 1890 - 1173 = 717

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